NOTE : Pressing Power button to turn off does NOT disconnect the appliance from the power supply.
Make sure that drainage is working properly. If water is not drained properly, your floor may get flooded. Flooded floors may induce electricity leakage, further resulting in electric shock or fire. If, during spinning, opening the lid does not stop the tub within about 15 seconds, immediately discontinue operating the machine. Call for repair. A machine that spins with its door open may cause injuries. Never reach into washer while it is moving. Wait until the machine has completely stopped before opening the lid. Even slow rotation can cause injury. Never attempt to operate this appliance if it is damaged, malfunctioning, partially disassembled, or has missing or broken parts, including a damaged cord or plug. Operating with a damaged plug may cause electric shock. Do not use water hotter than 50C. Use of excessively hot water may cause damage to fabrics or cause leakage of water.
Do not use a plug socket and wiring equipment for more than their rated capacity. Exceeding the limit may lead to electric shock, fire , break down, and/or deformation of parts. Do not mix chlorine bleach with ammonia or acids such as vinegar and/or rust remover. Mixing different chemicals can produce toxic gases which may cause death. Do not wash or dry articles that have been cleaned in, washed in, soaked in, or spotted with combustible or explosive substances (such as wax, oil, paint, gasoline, drycleaning solvents, kerosene, etc). Do not add these substances to the wash water. Do not use or place these substances around your washer or dryer during operation. It may ignite or explode.

WHEN NOT IN USE

Turn off water faucets to relieve pressure on hoses and valves and to minimize leakage if a break or rupture should occur. Check the condition of the fill hoses; they may need replacement after 5years. When the air temperature is high and the water temperature is low, condensation may occur and thus wet the floor. Wipe off dirt or dust on the contacts of the power plug. Using unclean power plug may cause fire. Before discarding a washer, or removing it from service, remove the washer lid to prevent children from hiding inside. Children may be trapped and suffocated if the lid is left intact. Do not attempt to repair or replace any part of this appliance unless specifically recommended in this Owners Manual, or in published user-repair instructions that you understand and have the skills to carry out. Operating the machine with improperly replaced parts may be a cause for fire or electric shock. Do not tamper with controls. It may lead to electric shock, fire , break down, deformation. Do not yank the power cord in an attempt to disconnect the power plug. Securely hold the power plug to unplug the machine. Failure to observe these instructions may cause electric shock or fire due to short-circuit. When you are cleaning the washing machine, do not apply water directly to any part of the washing machine. It will cause short circuit and electric shock.

dentification of Parts

DETERGENT BOX Detergent box for delayed washing / Softener box / Additives box COLD WATER SUPPLY HOSE Make sure the water does not leak. HOT WATER SUPPLY HOSE POWER PLUG If the supply cord is damaged, it must be replaced by the manufacturer or its service agents or similarly qualified person in order to avoid a hazard. FUNCTION SELECTOR

INLET HOLE FOR BLEACH

PULSATOR Vertical movement of washing action can be operated when a sufficient amount of laundry is deposited. Set the proper water level, an excessive amount of water may increase entanglement of laundry.
DRAIN HOSE Check if the drain hose is hung up before operating the washing machine.

MULTI CLEAN FILTER

LEVELLING LEGS Use to level the washing machine for correct balance & spin operation.
Accessories Inlet hose Drain hose Cable tie
1 Each for cold and hot water

Before Starting to Wash

reparation Before Washing

Care Labels

Look for a care label on your clothes. This will tell you about the fabric content of your garment and how it should be washed. Sort clothes into loads that can be washed with the same wash cycle, Water Temperature and spin speed.

Sorting

To get the best results, different fabrics need to be washed in different ways. SOIL (Heavy, Normal, Light) COLOR (Whites, Lights, Darks) LINT (Lint producers, Collectors) Lint Producers Lint Collectors Separate clothes according to the type and amount of soil. Separate white fabrics from colored fabrics. Separate lint producers and lint collectors. Terry cloth, Chenille, Towels, Nappies Synthetics, Corduroy, Permanent Press, Socks

Check before Loading

Check all pockets to make sure that they are empty. Things such as nails, hairclips, matches, pens, coins, and keys can damage both your washer and your clothes. Mend any torn garments or loose buttons. Tears or holes may become larger during washing. Remove belts, underwires, etc. to prevent damage to the machine or your clothes. Pretreat any dirt and stains. Make sure the clothes are washable in water. Check the washing instructions. Remove tissue in pockets.
Pretreatment on stains or heavy soil
Pretreat shirt collars and cuffs with a pre-wash product or liquid Detergent when placing them in the washer. Before washing treat special stains with bar soaps, liquid Detergent or a paste of water and granular Detergent. Use a pretreat soil and stain remover. Treat stains AS SOON AS POSSIBLE. The longer they are left the harder they are to remove. (For more detail refer to page 13)

Loading

Do not wash fabrics containing flammable materials (waxes, cleaning fluids, etc.). Load Size The WATER LEVEL should just cover the clothes. Adjust the load size accordingly. Loosely load clothes no higher than the top row of holes in the washer tub. To add items after washer has started, press START/PAUSE button and submerge additional items. Close the lid and press START/PAUSE button again to restart. Light and Large-sized clothing Clothes like downs and woollens are light weight, large and float easily. Use a nylon net and wash them in a small amount of water. If the laundry floats during the wash cycle, it may become damaged. Use dissolved Detergent to prevent the Detergent from clumping. Do not wash water-proof textilles (Suckling outfit, baby's nappy automobile seat covers.) Long laundry items Use nylon nets for long, DELICATES items. For laundry with long strings or long length, a net will prevent tangling during washing. Fasten zippers, hook, and strings to make sure that these items don't snag on other clothes. Nylon net is not supplied by LG.

WARNING

Fire Hazard
Never place items in the washer that are dampened with gasoline or other flammable fluids. No washer can completely remove oil. Do not dry anything that has ever had any type of oil on it (including cooking oils). Doing so can result in death, explosion, or fire.
se of Water, Detergent, Bleach, Softener, Additives
Using Water Amount of WATER LEVEL
This machine detects the quantity of laundry automatically, then sets the proper amount of water.

Water Temperature

The machine sets the appropriate temperature automatically according to the wash program. You can change the Water Temperature by pressing the WASH/RINSE button. The temperature of the water impacts the effectiveness of all laundry additives and therefore, the cleaning results. We recommend temperatures of: - HOT 49~60C.(120-140F) White items, diapers, underclothing and heavily soiled, colorfast items. - WARM 29~40C.(85-105F) Most items - COLD* 18~24C.(65-75F) Bright colors with light soil. When washing in COLD water additional steps may be needed: - Adjust Detergent amount and pre-dissolve Detergent in WARM water - Pretreat spots and stains - Soak heavily soiled items - Use appropriate bleach * Temperature below 18C.(65F) will not activate laundry additives and may cause lint, residue, poor cleaning, etc. In addition, Detergent manufactures and care labels define COLD water as 26~29C.(80-85F). If the temperature of the water in the tub is too cold for your hands, the Detergent will not activate and clean effectively. Note If iron is present in the water the clothes may become an all-over yellow or they may be stained with brown or orange spots or streaks. Iron is not always visible. Installation of water softener or an iron filter may be necessary for severe cases.

Using Detergent

Detergent
Follow the Detergent package directions. Using too little Detergent is a common cause of laundry problems. Use more Detergent if you have hard water, large loads, greasy or oily soils or lower Water Temperature.
Choosing the right Detergent
We recommend the use of drum type, low suding, high efficiency Detergent (powder, liquid or concentrated). Soap flakes or granulated soap powders should not be used in your washing machine. When washing woolens remember to use Detergent suitable for washing woolens.
Using the Liquid Bleach Dispenser
The bleach dispenser automatically dilutes and dispenses liquid chlorine bleach at the proper time in the wash cycle. 1. Check clothing care labels for special instructions. 2. Measure liquid bleach carefully, following instructions on the bottle. Never pour undiluted liquid chlorine bleach directly onto clothes or into the wash basket. Do not pour powdered bleach into bleach dispenser. Avoid overfilling or splashing when adding bleach to the dispenser. The maximum capacity of the bleach dispenser is one cup of bleach per wash cycle. Overfilling could result in early dispensing of bleach. 3. Before starting the washer, pour measured amount of bleach directly into bleach dispenser. If you prefer to use powdered bleach, add it into the wash basket directly before adding clothes.

Do not mix chlorine bleach with ammonia or acids such as vinegar and/or rust remover. Mixing can produce a toxic gas which may cause death The manufacturer's recommended amount of undiluted bleach goes into the bleach dispenser. During the final "Infusor" wash action the bleach is added to the wash load. This ensures performance won't be diminished. Two sequential flushe through the bleach dispenser completely removes the bleach from the dispenser. Any residual liquid left in the dispenser at the end of the cycle is water, not bleach. To prevent self-siphoning of the bleach into the wash basket and damage to your clothes, never add more than the maximum fill level marked on the dispenser. Also keep clothes away from the bleach dispenser so they don't absorb any bleach droplets left around the bleach dispenser.

WARNING!

Using the Dispenser Drawer
The dispenser drawer contains three compartments: Liquid Fabric Softener Powder Detergent Liquid Wash Boost Additives The dispenser automatically dispenses additives at the proper time in the wash cycle. 1. Slowly open the dispenser drawer by pulling the drawer out until it stops. 2. After adding laundry products, slowly close the dispenser drawer. Closing the drawer too quickly could result in early dispensing of additives. Avoid overfilling or splashing when adding laundry products to the dispenser. Doing so could result in early dispensing of aundry products. At the end of the cycle, you may see water in the compartments. This is part of the normal operation of the washer. NOTE: Do not use bleach in the dispenser drawer.

Adding Detergent

Add measured detergent to the detergent compartment of the dispenser drawer. Do not exceed the maximum fill line. Detergent is flushed through the dispenser at the beginning of the wash phase. Either powdered or liquid detergent can be used. Detergent usage may need to be adjusted for water temperature, water hardness, size and soil level of the load. Avoid using too much detergent in your washer, as it can lead to oversudsing and detergent residue being left on the clothes.

Adding Fabric Softener

If desired, pour the recommended amount of liquid fabric softener into the left-hand compartment. Use only liquid fabric softener. Dilute with water to the maximum fill line. Do not exceed the maximum fill line. Overfilling can cause early dispensing of the fabric which could stain clothes. NOTE: Do not pour fabric softener directly on the wash load. It may stain the clothes.
Adding Wash Boost Additives
The Wash Boost dispenser may be used to clean heavily soiled or stained garments more efficiently. Place the additives for the wash Boost setting in the righthand compartment. Using the Soak cycle add soaking detergent in the righthand compartment. Do not exceed the maximum fill line to avoid Overfilling can cause early dispensing of soak additives, which could result in damaged clothes.

unction of each Button

POWER BUTTON DELAY WASH (Reservation) BUTTON
Use to set a delayed finishing time. The time increases when the button is pushed. The following settings are indicated as the button is pushed 3 4. 16 18. 24 HOUR Reservation off 1. To cancel delay time, turn the power switch off or push DELAY WASH button until Reservation off. (refer to page 18)
Use to turn the power on or off. Push again and power goes on or off The power goes off automatically after the wash is finished. After turning the power on, if you dont push any button the power goes off automatically.

SOAK BUTTON

It gives additional Soak. The following settings are indicated as the button is pushed 120 180min. Soak off 15 (refer to page 19)

EXTRA RINSE BUTTON

It gives additional rinse cycle. (For better rinsing effect) (refer to page 20)

Custom Program BUTTON

This button allows you to store a customized wash cycle for future use. (refer to page 21)

SOIL LEVEL BUTTON

This button allows you to select the strength of the wash action. Power option selections light up in sequence as follows Normal Heavy Light Normal as the button is pushed. This can be selected for any program. Adjustment can be made while washing. Default SOIL LEVEL is Normal.

LOAD SENSING

It operates in all program. During being detected by the sensor, the LOAD SENSING LED flashes. It automatically sensing the quantity of laundries. During this step,the washing machine select optimized washing algorithm.
Note * For optimizing the washing algorithm, the displaying time may be changed.

SPIN SPEED

Spin speed can be selected from No Spin to Extra high. Default is Low. (refer to page 22)

BEEPER BUTTON

Beeper alerts you that the cycle is complete. The clothes should be removed when the beeper goes off so wrinkles won't set on. Beeper selection light up in sequence as follows: Low High Off Low Touch "Beeper" to select the volume or to turn the beeper off.

START/ PAUSE BUTTON

Use to start or pause the wash cycle. Changes can be made to the wash settings whilst the machine is paused. Repeats start and pause by pushing the button.

CYCLE BUTTON

Use for selecting wash program. This button allows you to select 6 different programs for different kinds of laundry and dirtiness. Program selections light up in sequence as follows: Fuzzy Heavy Duty Eco Cotton Wool/Silk Speed Wash Rinse/Spin Fuzzy etc. Select the desired program by pressing the button.

CHILD LOCK FUNCTION

Use to lock or unlock the control buttons to prevent settings from being changed by a child. To lock, push the SOIL LEVEL and WASH / RINSE buttons simultaneously during the wash mode and to unlock push them one more time during the washing process. (refer to page 23)

WASH / RINSE BUTTON

Use to select Water (Wash /Rinse) Temperature. Pressing the button allows you to select Cold/Cold Warm/Cold Hot/Cold respectively. Default setting is Warm/Cold.
* The standard detected by the sensor changing the algorithm is set by the normal standard. It may not be same to the washing habit of a specific user. So, it does not matter that the user uses it according to her/his washing habit.

ashing programs

Our machine provides various washing methods which suits various conditions and types of laundry

Function information

Add the laundry

Add the Detergent

Close the lid
This program is for mixed fabrics and automatically selects the most appropriate wash conditions such as wash action and cycle times by utilizing the built-in load sensor.

Heavy Duty

Use this Program for heavily soiled durable garments e.g overalls, jeans.

Eco Cotton

Eco Cotton is the program for normally soiled cotton material and designed to minimize water consumption.

Wool / Silk

Use this program for washing delicate fabrics such as lingerie and woolens.(wash only "Water washable" clothes) Before washing your woolens check the care label for the washing instructions. (Wash only water washable clothes)

Speed Wash

Use this program for lightly soiled garments.

Rinse/Spin

When you only need the Rinse/Spin cycle, these can be set manually.
Select the program on the Control panel.

Finish

Caution & Note
Usage of Water is already optimized to every level.
Thick and heavy clothes or those which are excessively dirty like jeans or working uniforms can be washed.
Delicates clothes (lingerie, wool, etc) which may be easily damaged can be washed. The fibers of machine washable woolens have been specifically modified to prevent shrink when they are machine washed. Most hand knitted garments are not made of machine washable wool and we recommend that you hand wash them. It reduce the washing time.

elay Wash (Reservation)

Delay Wash (Reservation) is used to delay the finishing time of the operation. The hours to be delayed can be set by the user accordingly. The time on the display is the finishing time, not the start time.

Add the laundry into the washing tub. Add the Detergent Close the lid
When the lid is open the machine will not operate, and an alarm Signal or will remind you to close the lid when it starts.
Press the POWER button to turn power on.
Press the CYCLE button to select the washing program.
Select desired Options. SOIL LEVLE, WASH/RINSE, SPIN SPEED, BEEPER, SOAK, EXTRA RINSE Press the Delay Wash (Reservation) button.
The light of 'Delay Wash' will be turned on and TIME LEFT will be marked. Press the button repeatedly to set the desired finishing time. For example, To finish washing in 9 hours from now, by make the number 9:00 pressing the Delay Wash (Reservation) button repeatedly.

(RES.=Delay)

Press the START/PAUSE button.
When you press the START/PAUSE button the light will blink.
The washing will be finished according to the delayed time.
Finishing time can be delayed from 1~24 hours. Delaying from 1~12 hours can be done in 1 hour time intervals and from 12~24 hours in 2 hour time intervals. If the lid is open, the machine will not work, and an alarm Signal or will remind you to close the lid. To select Spin Speed, water temp. or soil level manually, press the Delay Wash (Reservation) button and select the desired option. Then press the START/PAUSE button. When the laundry cannot be taken out immediately after the wash program ends, it is better to omit the spinning program. (The laundry will be wrinkled if left for a long time after spinning.)

oak Wash

Use this mode to wash normal clothes or thick and heavy clothes which are excessively dirty. Soak mode can be used all program except Rinse/Spin.
Add the laundry into the washing tub. Selezionare il programma preferito. Add the detergent in the right-hand compartment for Soak Wash cycle and in the middle compartment for main wash cycle.
Press the CYCLE button to select the washing program. Select the cycle for laundry on the Control Panel.
Select desired Options. SOIL LEVLE, WASH/RINSE, SPIN SPEED, BEEPER, EXTRA RINSE Press the Soak button to change the soaking time. The following settings are indicated as the button is pushed. 120 180min SOAK OFF Press the START/PAUSE button.
The Soak wash cycle run prior to the main wash cycle. For some cycles,the water temperature for the soak cycle may not be the same as the water temperature for the main wash cycle.

xtra Rinse

This Program gives additional rinse cycle.
Select desired Options. SOIL LEVLE, WASH/RINSE, SPIN SPEED, BEEPER, SOAK, DELAY WASH
Press the Extra Rinse button to give additional rinse cycle.
Note This CYCLE is for better rinsing effect.

ustom Program

Custom Program allows you to store a customized wash cycle for future use.
Select desired Options. SOIL LEVLE, WASH/RINSE, SPIN SPEED, BEEPER, SOAK, EXTRA RINSE, DELAY WASH
Press the Custom Program button for 3 seconds to store your Custom wash cycle.
Note To recall your Custom cycle. 1. Press POWER on. 2. Select the Custom cycle by pressing the Custom Program button. 3. Press the START/PAUSE button. Then the wash cycle starts.

pin Only

When you only need the Spin cycles,this can be set manually.
Add the laundry into the washing tub.
Press the SPIN SPEED button. The following settings are indicated as the button is pushed Low Medium High Extra High Low.
Note Fill laundry uniformly in the inner tub

hild Lock

If you want to lock all the keys to prevent settings from being changed during the wash operation by a child, you can use the child lock option.

How to Lock

Press the POWER button

Turn Power on.

After all washing conditions are set according to the manual.
Press both the SOIL LEVEL button and the WASH/RINSE button simultaneously.
During the wash program,all the buttons are locked until washing is completed or it is child-lock function is deactivated manually.

How to Unlock

If you want to unlock during wash, press both the SOIL LEVEL and the WASH/RINSE button simultaneously again.
Note & the remaining time are alternatively shown on the display while they are locked.

lacing and Leveling

Please ensure that transit restraint has been removed before operating your machine.
Place the machine on a flat and firm surface which allows proper clearance.
We suggest a minimum clearance of 2 cm to the right side and left sides of the machine for ease of installation. At the rear, a minimum clearance of 10 cm is suggested.
more than 10 cm more than 2 cm more than 2 cm
Caution If the washer is installed on a uneven, weak or tilted floor causing excessive vibration, spin failure or " " error can happen to it. It must be placed on a firm and level floor to prevent spin failure.

Set rear legs

Remove the level and tilt the washer forward (pivot on front legs) about 4 to 6 inches and gently lower the rear legs back onto the floor. This action will set rear leg adjustments to correspond to front settings.

0.9~1.2m

Cable Tie
Note The discharge height should be approximately 0.9~1.2m from the floor. Do not lay down the drain hose during wash mode.

rounding Method

Care and Maintenance
Earth wire should be connected. If the earth wire is not connected, there is possible a danger of electric shock caused by the current leakage.
Grounding Method with Ground insert space Terminal
If the AC current outlet has a ground terminal, then separate grounding is not required. Note that AC power outlet configurations may differ from country to country.
Caution CAUTION concerning the Power Cord Most appliances recommend they be placed upon a dedicated circuit; that is, a single outlet circuit which powers only that appliance and has no additional outlets or branch circuits. Check the spercification page of this owner's manual to be certain. Do not overload wall outlets. Overloaded wall outlets, loose or damaged wall outlets, extension cords, frayed power cords, or damaged or cracked wire insulation are dangerous. Any of theseconditions could result in electric shock or fire. Periodically examine the cord of your appliance, and if its appearance indicates damage or deterioration, unplug it, discontinue use of the appliance, and have the cord replaced with an exact replacement part by an authorized servicer. Protect the power cord form physical or machanical abuse, such as being twisted, kinked, pinched, colsed in a door, or walked upon. Pay particular attention to plugs. wall outlets, and the point where the cord exits the appliance.

Other Grounding Method

Burying Copper Plate
Connect the Ground Wire to a Ground Copper Plate and bury it more than 75cm in the ground.

Ground Wire

Ground Copper Plate

Using Ground Wire

Connect the Ground Wire to a the socket provided exclusively for grounding
Using a Short Circuit Breaker
If grounding methods described above are not possible, a separate circuit breaker should be employed and installed by a qualified electrician Caution To prevent a possible explosion, do not connect ground to a gas pipe. Do not connect ground to telephone wires or lightning rods. This may be dangerous during electrical storms. Connecting ground to plastic has no effect. Ground wires should be connected when an extension cord is used.

Short-circuit breaker

leaning and Maintenance

Cold water washing

If you always use COLD water, we recommend that a WARM or HOT wash be used at a regular intervals e.g. every 5th wash should be at least a WARM one.

When you have finished

Turn off taps to prevent the chance of flooding should a hose burst. Always unplug the power cord after use. When water supply into the tub is not clean or the filter is clogged with particles (such as sand), clean the filter in the inlet valve occasionally. (The figure of power cord and water tap may vary according to the country)
To Clean the Filter in the Inlet Valve
Close the tap before turning off the power. Select both Hot/Cold and then press the [START/PAUSE] button to remove water from the machine completely. After removing the water supply hose pull out the filter. Then use a brush to clean the filter. Caution
Turn off the power before pulling out the cord.
Replace the filter after cleaning it.
Before cleaning the filter, the impurities in the water supply hose should be removed.
How to clean Multi clean Filter
Push the upper part of multi-clean filter down and pull forward.
Open the lid and remove lint and then wash out.
Close the lid and at first get it back to tub. First insert the lower part of filter to washer tub and pull it down until a click sounds.
Without multi clean filter, laundry may get damaged. Clean multi clean filter as often as possible. (Foreign objects accumulation may cause smell and filtering performance will be worsened. In order to reduce lint generation, - laundry which creates lint easily should get turned inside out. - Wash load sorted by color.
When there is a fear of freezing
Close the water taps and remove the Water Supply Hose. Remove the water which remains in the water supply. Lower the drain hose and drain the water in the bowl and the drain hose by spinning.

If frozen

Remove the water supply hose, and immerse it in HOT water at approx.40 Pour approx.2 liters of HOT water at approx.40C, into the bowl and let it stand for 10 minutes. Connect the water supply hose to the water tap and confirm that the washing machine performs the supply and drainage of water. Wash Inner-Tub Leave the lid open after washing to allow moisture to evaporate. If you want to clean the inner-tub use a clean soft cloth dampened with liquid Detergent, then rinse. (Do not use harsh or gritty cleaners.) Hoses connecting washer to faucet should be replaced every 5 years. Immediately wipe off any spills. Wipe with damp cloth. Try not to hit surface with sharp objects. Be sure water supply is shut off at faucets. Drain all water from hoses if weather will be below freezing.

Inlet Hoses Exterior Long Vacations
Cleaning the Inside of your Washer
If you use fabric softener or do regular COLD water washing, it is very important that you occasionally clean the inside of your washer. Fill your washer with HOT water. Add about two cups of a powdered Detergent that contains phosphate. Let it operate for several minutes. Stop the washer, open the lid and leave it to soak overnight. After soaking, drain the washer and run it through a normal cycle.
Products that might damage your washing machine
Concentrated bleaches and diaper sanitizer will cause damage to the paintwork and components of your washer. Hydrocarbon solvents i.e. petrol, paint thinners and lacquer thinners, etc. can dissolve plastic and blister paint (Be careful when washing garments stained with these solvents as they are flammable DO NOT put them in washer or dryer.) Some pretreatment sprays or liquids can damage your washers control panel. Use of dyes in your washer may cause staining of the plastic components. The dye will not damage the machine but we suggest you thoroughly clean your washer afterwards. We do not recommend the use of dye strippers in your washer. Do not use your washer lid as a work surface.

Troubleshooting

ommon washing problems
Many washing problems involve poor soil & stain removal, residues of lint and scum, and fabric damage. For satisfactory washing results, follow these instructions.

WASH PROBLEM

Problems Poor soil removal Possible Causes
Not enough Detergent Wash Water Temperature
Solutions& Preventive Measures
Use correct amount of Detergent for load size,
amount of soil and water Hardness.
Use WARM or HOT water for normal soil. Different
too low. Overloading the washer Incorrect wash cycle Incorrect sorting Do not pretreat stain.
Water Temperature may be required according to soil type. (refer to page 10) Reduce load size. Wash with Heavy or Soak & Heavy wash cycle for heavy soiled laundry. Separate heavily soiled items from lightly soiled ones. Pretreat stain and heavy soil according to directions shown on page 9.

Blue Stains

Undiluted fabric softener
dispensed directly onto fabric
Rub the stain with bar soap Wash. Do not overfill fabric softener dispenser and do not
pour liquid fabric softener directly onto fabric. See page 12 for more instructions.
Black or gray marks on clothes

A buildup caused by the

Keep the recommendations against Scrud(waxy interaction of fabric buildup). softener and Detergent can Use correct amount of Detergent for load size, amount of soil and water Hardness. flake off and mark clothes Not enough Detergent To restore discolored load of whites, use rust

What Is Not Covered:

Service trips to your home to teach you how to use the product. If the product is connected to any voltage other than that shown on the rating plate. If the fault is caused by accident, neglect, misuse or Act of God. If the fault is caused by factors other than normal domestic use or use in accordance with the owner's manual. Provide instruction on use of product or change the set-up of the product. If the fault is caused by pests for example, rats or cockroaches etc. Noise or vibration that is considered normal for example water drain sound, spin sound, or warming beeps. Correcting the installation for example, levelling the product, adjustment of drain. Normal maintenance which recommended by the owner's manual. Removal of foreign objects / substances from the machine, including the pump and inlet hose filter for example, grit, nails, bra wires, buttons etc. Replace fuses in or correct house wiring or correct house plumbing. Correction of unauthorized repairs. Incidental or consequential damage to personal property caused by possible defects with this appliance. If this product is used for commercial purpose, it is not warranted. (Example : Public places such as public bathroom, lodging house, training center, dormitory
If the product is installed outside the normal service area, any cost of transportation involved in the repair of the product, or the replacement of a defective part, shall be borne by the owner.

pecification

Specification
Model Size Weight Water tap pressure Wash Capacity
WT-R854 685(W) X 720(D) X 1095(H) 58kg 30-600 kPa(0.3-6.0 kgf/cm2) 8.5kg
In our continuing effort to improve the quality of our appliances, it may be necessary to make changes to the appliance without revising this manual.

List of Figures 4.5 4.6 5.1 5.2 5.3 7.1
Deleting the root of a trivial binary heap. 146 Delete and sift-down in a binary heap. 147 A claw graph with 4 vertices. 157 The Petersen graph with 10 vertices. 159 A digraph with 6 vertices. 162 The Errera graph is planar. 171

List of Algorithms

1.1 1.2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 4.1 4.2 4.3 Computing graph isomorphism using canonical labels. Havel-Hakimi test for sequences realizable by simple graphs. A general breadth-rst search template. A general depth-rst search template. Determining whether an undirected graph is connected. A template for shortest path algorithms. A general template for Dijkstras algorithm. The Bellman-Ford algorithm. The Bellman-Ford algorithm with checks for redundant updates. The Floyd-Roy-Warshall algorithm for all-pairs shortest paths. Variant of the Floyd-Roy-Warshall algorithm for transitive closure. Johnsons algorithm for sparse graphs. The Euclidean algorithm. Polynomial evaluation using Horners method. Linear search for lists. Binary search for lists of positive integers. Bubble sort. Swapping values using a temporary placeholder. Selection sort. A brackets parser. Randomized spanning tree construction. Kruskals algorithm. Prims algorithm. Borvkas algorithm. u Random binary tree. Binary tree representation of Human codes. Human encoding of an alphabet. Level-order traversal. Pre-order traversal. Post-order traversal. Bottom-up traversal. In-order traversal. Random spanning tree of Kn. 133
Sorting a sequence via priority queue. 139 Inserting a new internal vertex into a binary heap. 146 Extract the minimum vertex of a binary heap. 148 viii
List of Algorithms 4.4 5.1 5.2
Heapify a binary tree. 149 Ford-Fulkerson algorithm. 163 Friendship graph. 167
10.1 Random simple, directed, acyclic, or weighted graph. 177

List of Tables

2.1 2.2 2.3 2.4 2.5 2.6 3.1 ASCII printable characters used by graph6 and sparse6. Big-endian order of the ASCII binary code of E. Little-endian order of the ASCII binary code of E. Stepping through Dijkstras algorithm. Implementation specic worst case time complexity of Dijkstras algorithm. Distances in kilometers between major world capital cities. 68 85

Morse code. 118

A.1 Meaning of asymptotic notations. 192 A.2 Asymptotic behavior in the limit of large n. 192
Chapter 1 Introduction to Graph Theory
Spiked Math, http://spikedmath.com/120.html Our journey into graph theory starts with a puzzle that was solved over 250 years ago by Leonhard Euler (17071783). The Pregel River owed through the town of Knigsberg, o which is present day Kaliningrad in Russia. Two islands protruded from the river. On either side of the mainland, two bridges joined one side of the mainland with one island and a third bridge joined the same side of the mainland with the other island. A bridge connected the two islands. In total, seven bridges connected the two islands with both sides of the mainland. A popular exercise among the citizens of Knigsberg was o determining if it was possible to cross each bridge exactly once during a single walk. For historical perspectives on this puzzle and Eulers solution, see Gribkovskaia et al. [41] and Hopkins and Wilson [50]. To visualize this puzzle in a slightly dierent way, consider Figure 1.1. Imagine that points a and c are either sides of the mainland, with points b and d being the two islands. Place the tip of your pencil on any of the points a, b, c, d. Can you trace all the lines in the gure exactly once, without lifting your pencil? Known as the seven bridges of Knigsberg puzzle, Euler solved this problem in 1735 and with his solution he laid the o foundation of what is now known as graph theory.

sage : G = Graph ({1:[2 ,4] , 2:[1 ,4] , 3:[2 ,6] , 4:[1 ,3] , 5:[4 ,2] , 6:[3 ,1]}) sage : d = [[ G. distance (i , j ) for i in range (1 ,7)] for j in range (1 ,7)] sage : matrix ( d ) [2 1] [1 2] [2 1] [1 2] [0 3] [3 0]
The distance matrix is an important quantity which allows one to better understand the connectivity of a graph. Distance and connectivity will be discussed in more detail in Chapters 5 and 10.

Isomorphic graphs

Determining whether or not two graphs are, in some sense, the same is a hard but important problem. Two graphs G and H are isomorphic if there is a bijection f : V (G) V (H) such that whenever uv E(G) then f (u)f (v) E(H). The function f is an isomorphism between G and H. Otherwise, G and H are non-isomorphic. If G and H are isomorphic, we write G H. =
Figure 1.17: Two representations of the Franklin graph.

e f e f

a (a) C6

5 (b) G1

a (c) G2
Figure 1.18: Isomorphic and nonisomorphic graphs. A graph G is isomorphic to a graph H if these two graphs can be labelled in such a way that if u and v are adjacent in G, then their counterparts in V (H) are also adjacent in H. To determine whether or not two graphs are isomorphic is to determine if they are structurally equivalent. Graphs G and H may be drawn dierently so that they seem dierent. However, if G H then the isomorphism f : V (G) V (H) shows that both = of these graphs are fundamentally the same. In particular, the order and size of G are equal to those of H, the isomorphism f preserves adjacencies, and deg(v) = deg(f (v)) for all v G. Since f preserves adjacencies, then adjacencies along a given geodesic path are preserved as well. That is, if v1 , v2 , v3 ,. , vk is a shortest path between v1 , vk V (G), then f (v1 ), f (v2 ), f (v3 ),. , f (vk ) is a geodesic path between f (v1 ), f (vk ) V (H). For example, the two graphs in Figure 1.17 are isomorphic to each other. Example 1.21. Consider the graphs in Figure 1.18. Which pair of graphs are isomorphic, and which two graphs are non-isomorphic? Solution. If G is a Sage graph, one can use the method G.is_isomorphic() to determine whether or not the graph G is isomorphic to another graph. The following Sage session illustrates how to use G.is_isomorphic().
sage :. sage :. sage :. sage : True sage : False C6 = Graph ({ " a " :[ " b " ," c " ] , " b " :[ " a " ," d " ] , " c " :[ " a " ," e " ] , \ " d " :[ " b " ," f " ] , " e " :[ " c " ," f " ] , " f " :[ " d " ," e " ]}) G1 = Graph ({1:[2 ,4] , 2:[1 ,3] , 3:[2 ,6] , 4:[1 ,5] , \ 5:[4 ,6] , 6:[3 ,5]}) G2 = Graph ({ " a " :[ " d " ," e " ] , " b " :[ " c " ," f " ] , " c " :[ " b " ," f " ] , \ " d " :[ " a " ," e " ] , " e " :[ " a " ," d " ] , " f " :[ " b " ," c " ]}) C6. is_isomorphic ( G1 ) C6. is_isomorphic ( G2 )

Figure 1.24 shows a sequence of graphs resulting from edge deletion. Unlike vertex deletion, when an edge is deleted the vertices incident on that edge are left intact.

c (a) G

c (b) G {ac}

c (c) G {ab, ac, bc}

Figure 1.24: Obtaining subgraphs via repeated edge deletion.
Vertex cut, cut vertex, or cutpoint A vertex cut (or separating set) of a connected graph G = (V, E) is a subset W V such that the vertex deletion subgraph G W is disconnected. In fact, if v1 , v2 V are two non-adjacent vertices, then you can ask for a vertex cut W for which v1 , v2 belong to dierent components of G W. Sages vertex_cut method allows you to compute a minimal cut having this property. For many connected graphs, the removal of a single vertex is sucient for the graph to be disconnected (see Figure 1.24(c)). Edge cut, cut edge, or bridge If deleting a single, specic edge would disconnect a graph G, that edge is called a bridge. More generally, the edge cut (or disconnecting set or seg) of a connected graph G = (V, E) is a set of edges F E whose removal yields an edge deletion subgraph G F that is disconnected. A minimal edge cut is called a cut set or a bond. In fact, if v1 , v2 V are two vertices, then you can ask for an edge cut F for which v1 , v2 belong to dierent components of G F. Sages edge_cut method allows you to compute a minimal cut having this property. For example, any of the three edges in Figure 1.24(c) qualies as a bridge and those three edges form an edge cut for the graph in question. Theorem 1.25. Let G be a connected graph. An edge e E(G) is a bridge of G if and only if e does not lie on a cycle of G. Proof. First, assume that e = uv is a bridge of G. Suppose for contradiction that e lies on a cycle C : u, v, w1 , w2 ,. , wk , u. Then G e contains a u-v path u, wk ,. , w2 , w1 , v. Let u1 , v1 be any two vertices in G e. By hypothesis, G is connected so there is a u1 -v1 path P in G. If e does not lie
on P , then P is also a path in G e so that u1 , v1 are connected, which contradicts our assumption of e being a bridge. On the other hand, if e lies on P , then express P as u1 ,. , u, v,. , v1 Now u1 ,. , u, wk ,. , w2 , w1 , v,. , v1 or u1 ,. , v, w1 , w2 ,. , wk , u,. , v1 or u1 ,. , v, u,. , v1.
respectively is a u1 -v1 walk in G e. By Theorem 1.13, G e contains a u1 -v1 path, which contradicts our assumption about e being a bridge. Conversely, let e = uv be an edge that does not lie on any cycles of G. If G e has no u-v paths, then we are done. Otherwise, assume for contradiction that G e has a u-v path P. Then P with uv produces a cycle in G. This cycle contains e, in contradiction of our assumption that e does not lie on any cycles of G. Edge contraction An edge contraction is an operation which, like edge deletion, removes an edge from a graph. However, unlike edge deletion, edge contraction also merges together the two vertices the edge used to connect. For a graph G = (V, E) and an edge uv = e E, the edge contraction G/e is the graph obtained as follows: 1. Delete the vertices u, v from G. 2. In place of u, v is a new vertex ve. 3. The vertex ve is adjacent to vertices that were adjacent to u, v, or both u and v. The vertex set of G/e = (V , E ) is dened as V = V \{u, v} {ve } and its edge set is E = wx E | {w, x} {u, v} = ve w | uw E\{e} or vw E\{e}. Make the substitutions E1 = wx E | {w, x} {u, v} = E2 = ve w | uw E\{e} or vw E\{e}. Let G be the wheel graph W6 in Figure 1.25(a) and consider the edge contraction G/ab, where ab is the gray colored edge in that gure. Then the edge set E1 denotes all those edges in G each of which is not incident on a, b, or both a and b. These are precisely those edges that are colored red. The edge set E2 means that we consider those edges in G each of which is incident on exactly one of a or b, but not both. The blue colored edges in Figure 1.25(a) are precisely those edges that E2 suggests for consideration. The result of the edge contraction G/ab is the wheel graph W5 in Figure 1.25(b). Figures 1.25(a) to 1.25(f) present a sequence of edge contractions that starts with W6 and repeatedly contracts it to the trivial graph K1.

1.7. Problems 1.12. Show that the complement of an edgeless graph is a complete graph.
1.13. Let G H be the Cartesian product of two graphs G and H. Show that |E(G H)| = |V (G)| |E(H)| + |E(G)| |V (H)|.
Figure 1.34: Eulers polygon division problem for the hexagon. 1.14. In 1751, Leonhard Euler posed a problem to Christian Goldbach, a problem that now bears the name Eulers polygon division problem. Given a plane convex polygon having n sides, how many ways are there to divide the polygon into triangles using only diagonals? For our purposes, we consider only regular polygons having n sides for n 3 and any two diagonals must not cross each other. For example, the triangle is a regular 3-gon, the square a regular 4-gon, the pentagon a regular 5-gon, etc. In the case of the hexagon considered as the cycle graph C6 , there are 14 ways to divide it into triangles, as shown in Figure 1.34, resulting in 14 graphs. However, of those 14 graphs only 3 are nonisomorphic to each other. (a) What is the number of ways to divide a pentagon into triangles using only diagonals? List all such divisions. If each of the resulting so divided pentagons is considered a graph, how many of those graphs are nonisomorphic to each other? (b) Repeat the above exercise for the heptagon. (c) Let En be the number of ways to divide an n-gon into triangles using only diagonals. For n 1, the Catalan numbers Cn are dened as Cn = 2n 1. n+1 n
Drrie [25, pp.2127] showed that En is related to the Catalan numbers via o the equation En = Cn1. Show that Cn = 1 2n + 2. 4n + 2 n + 1
Chapter 1. Introduction to Graph Theory For k 2, show that the Catalan numbers satisfy the recurrence relation Ck = 4k 2 Ck1. k+1
1.15. A graph is said to be planar if it can be drawn on the plane in such a way that no two edges cross each other. For example, the complete graph Kn is planar for n = 1, 2, 3, 4, but K5 is not planar (see Figure 1.12). (a) Draw a planar version of K4 as presented in Figure 1.12(b). (b) Is the graph in Figure 1.8 planar? (c) For n = 1, 2,. , 5, enumerate all simple nonisomorphic graphs on n vertices that are planar. For simplicity, only work with undirected graphs. 1.16. If n 3, show that the join of Cn and K1 is the wheel graph Wn+1. In other words, show that Cn + K1 = Wn+1. 1.17. A common technique for generating random numbers is the linear congruential method, a generalization of the Lehmer generator [71] introduced in 1949. First, we choose four integers: m, a, c, modulus, multiplier, 0<m 0a<m 0 X0 < m

increment, 0 c < m

X0 , seed,
where the value X0 is also referred to as the starting value. Then iterate the relation Xn+1 = (aXn + c) mod m, n0 and halt when the relation produces the seed X0 or when it produces an integer Xk such that Xk = Xi for some 0 i < k. The resulting sequence S = (X0 , X1 ,. , Xn ) is called a linear congruential sequence. Dene a graph theoretic representation of S as follows: let the vertex set be V = {X0 , X1 ,. , Xn } and let the edge set be E = {Xi Xi+1 | 0 i < n}. The resulting graph G = (V, E) is called the linear congruential graph of the linear congruential sequence S. See chapter 3 of Knuth [61] for other techniques for generating random numbers. (a) Compute the linear congruential sequences Si with the following parameters: (i) (ii) (iii) (iv) S1 : S2 : S3 : S4 : m = 10, m = 10, m = 10, m = 10, a = c = X0 = 7 a = 5, c = 7, X0 = 0 a = 3, c = 7, X0 = 2 a = 2, c = 5, X0 = 3

Adjacency lists

A list is a sequence of objects. Unlike sets, a list may contain multiple copies of the same object. Each object in a list is referred to as an element of the list. A list L of n 0 elements is written as L = [a1 , a2 ,. , an ], where the i-th element ai can be indexed as L[i]. In case n = 0, the list L = [ ] is referred to as the empty list. Two lists are equivalent if they both contain the same elements at exactly the same positions. Dene the adjacency lists of a graph as follows. Let G be a graph with vertex set V = {v1 , v2 ,. , vn }. Assign to each vertex vi a list Li containing all the vertices that are adjacent to vi. The list Li associated with vi is referred to as the adjacency list of vi. Then Li = [ ] if and only if vi is an isolated vertex. We say that Li is the adjacency list of vi because any permutation of the elements of Li results in a list that contains the same vertices adjacent to vi. We are mainly concerned with the neighbors of vi , but disregard the position where each neighbor is located in Li. If each adjacency list Li contains si elements where 0 si n, we say that Li has length si. The adjacency list representation of the graph G requires that we represent i si = 2 |E(G)| n2 elements in a computers memory, since each edge appears twice in the adjacency list representation. An adjacency list is explicit about which vertices are adjacent to a vertex and implicit about which vertices are not adjacent to that same vertex. Without knowing the graph G, given the adjacency lists L1 , L2 ,. , Ln , we can reconstruct G. For example, Figure 2.1 shows a graph and its adjacency list representation.
L1 = [2, 8] L2 = [1, 6] L3 = [4]
L5 = [6, 8] L6 = [2, 5, 8] L7 = [ ] L8 = [1, 5, 6]

L4 = [3]

Figure 2.1: A graph and its adjacency lists. Example 2.1. The Kneser graph with parameters (n, k), also known as the (n, k)-Kneser graph, is the graph whose vertices are all the k-subsets of {1, 2,. , n}. Furthermore, two vertices are adjacent if their corresponding sets are disjoint. Draw the (5, 2)-Kneser graph and nd its order and adjacency lists. In general, if n and k are positive, what is the order of the (n, k)-Kneser graph?
Chapter 2. Graph Algorithms
Solution. The (5, 2)-Kneser graph is the graph whose vertices are the 2-subsets {1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 5} of {1, 2, 3, 4, 5}. That is, each vertex of the (5, 2)-Kneser graph is a 2-combination of the set {1, 2, 3, 4, 5} and therefore the graph itself has order 5 = 54 = 10. The edges of 2! 2 this graph are ({1, 3}, {2, 4}), ({2, 4}, {1, 5}), ({2, 4}, {3, 5}), ({1, 3}, {4, 5}), ({1, 3}, {2, 5}) ({3, 5}, {1, 4}), ({3, 5}, {1, 2}), ({1, 4}, {2, 3}), ({1, 4}, {2, 5}), ({4, 5}, {2, 3}) ({4, 5}, {1, 2}), ({1, 5}, {2, 3}), ({1, 5}, {3, 4}), ({3, 4}, {1, 2}), ({3, 4}, {2, 5}) from which we obtain the following adjacency lists: L{1,2} = [{3, 4}, {3, 5}, {4, 5}], L{1,4} = [{2, 3}, {3, 5}, {2, 5}], L{2,3} = [{1, 5}, {1, 4}, {4, 5}], L{2,5} = [{1, 3}, {3, 4}, {1, 4}], L{3,5} = [{2, 4}, {1, 2}, {1, 4}], L{1,3} = [{2, 4}, {2, 5}, {4, 5}], L{1,5} = [{2, 4}, {3, 4}, {2, 3}], L{2,4} = [{1, 3}, {1, 5}, {3, 5}], L{3,4} = [{1, 2}, {1, 5}, {2, 5}], L{4,5} = [{1, 3}, {1, 2}, {2, 3}].

0 (a) Original undirected graph. 0 (c) 2nd iteration of while loop. 0 (e) 4th iteration of while loop. 11
(b) 1st iteration of while loop. 6
(d) 3rd iteration of while loop. 6

(f) Final MST.

Figure 3.13: Running Prims algorithm over an undirected graph.
(a) Original undirected graph. 2 10
(b) 0th iteration of while loop. 10
(c) 1st iteration of while loop.
(d) 2nd iteration of while loop.
Figure 3.14: Recursive construction of MST via Borvkas algorithm. u
to , or None if x is not in flatten ( L ). The 0 - th element in Lx = [ L. index ( S ) for S in L if x in S ] almost works , but if the list is empty then Lx [0] throws an exception. EXAMPLES : sage : L = [[1 ,2 ,3] ,[4 ,5] ,[6 ,7 ,8]] sage : which_index (3 , L ) 0 sage : which_index (4 , L ) 1 sage : which_index (7 , L ) 2 sage : which_index (9 , L ) sage : which_index (9 , L ) == None True """ for S in L : if x in S : return L. index ( S ) return None def boruvka ( G ): """ Implements Boruvka s algorithm to compute a MST of a graph. INPUT : G - a connected edge - weighted graph with distinct weights. OUTPUT : T - a minimum weight spanning tree.
REFERENCES : http :// en. wikipedia. org / wiki / Boruvka s_algorithm """ T_vertices = [] # assumes G. vertices = range ( n ) T_edges = [] T = Graph () E = G. edges () # a list of triples V = G. vertices () # start ugly hack to sort E Er = [ list ( x ) for x in E ] E0 = [] for x in Er : x. reverse () E0. append ( x ) E0. sort () E = [] for x in E0 : x. reverse () E. append ( tuple ( x )) # end ugly hack to get E is sorted by weight for e in E : # create about | V |/2 edges of T " cheaply " TV = T. vertices () if not ( e [0] in TV ) or not ( e [1] in TV ): T. add_edge ( e ) for e in E : # connect the " cheapest " components to get T C = T. c o n n e c t e d _ c o m p o n e n t s _ s u b g r a p h s () VC = [ S. vertices () for S in C ] if not ( e in T. edges ()) and ( which_index ( e [0] , VC ) != which_index ( e [1] , VC )): if T. is_connected (): break T. add_edge ( e ) return T
Some examples using Sage:
sage : A = matrix ([[0 ,1 ,2 ,3] , [4 ,0 ,5 ,6] , [7 ,8 ,0 ,9] , [10 ,11 ,12 ,0]]) sage : G = DiGraph (A , format = " adjacency_matrix " , weighted = True ) sage : boruvka ( G ) Multi - graph on 4 vertices sage : boruvka ( G ). edges () [(0 , 1 , 1) , (0 , 2 , 2) , (0 , 3 , 3)] sage : A = matrix ([[0 ,2 ,0 ,5 ,0 ,0 ,0] , [0 ,0 ,8 ,9 ,7 ,0 ,0] , [0 ,0 ,0 ,0 ,1 ,0 ,0] ,\
. [0 ,0 ,0 ,0 ,15 ,6 ,0] , [0 ,0 ,0 ,0 ,0 ,3 ,4] , [0 ,0 ,0 ,0 ,0 ,0 ,11] , [0 ,0 ,0 ,0 ,0 ,0 ,0]]) sage : G = Graph (A , format = " adjacency_matrix " , weighted = True ) sage : E = G. edges (); E [(0 , 1 , 2) , (0 , 3 , 5) , (1 , 2 , 8) , (1 , 3 , 9) , (1 , 4 , 7) , (2 , 4 , 1) , (3 , 4 , 15) , (3 , 5 , 6) , (4 , 5 , 3) , (4 ,6 , 4) , (5 , 6 , 11)] sage : boruvka ( G ) Multi - graph on 7 vertices sage : boruvka ( G ). edges () [(0 , 1 , 2) , (0 , 3 , 5) , (2 , 4 , 1) , (3 , 5 , 6) , (4 , 5 , 3) , (4 , 6 , 4)] sage : A = matrix ([[0 ,1 ,2 ,5] , [0 ,0 ,3 ,6] , [0 ,0 ,0 ,4] , [0 ,0 ,0 ,0]]) sage : G = Graph (A , format = " adjacency_matrix " , weighted = True ) sage : boruvka ( G ). edges () [(0 , 1 , 1) , (0 , 2 , 2) , (2 , 3 , 4)] sage : A = matrix ([[0 ,1 ,5 ,0 ,4] , [0 ,0 ,0 ,0 ,3] , [0 ,0 ,0 ,2 ,0] , [0 ,0 ,0 ,0 ,0] , [0 ,0 ,0 ,0 ,0]]) sage : G = Graph (A , format = " adjacency_matrix " , weighted = True ) sage : boruvka ( G ). edges () [(0 , 1 , 1) , (0 , 2 , 5) , (1 , 4 , 3) , (2 , 3 , 2)]

Tree representation

Any binary code can be represented by a tree, as Example 3.19 shows. Example 3.19. Let B be the binary code of length . Represent codewords of B using trees. Solution. Here is how to represent the code B consisting of all binary strings of length . Start with the root node being the empty string. The two children of this node, v0 and v1 , correspond to the two strings of length 1. Label v0 with a 0 and v1 with a 1. The two children of v0 , i.e. v00 and v01 , correspond to the strings of length 2 which start with a 0. Similarly, the two children of v1 , i.e. v10 and v11 , correspond to the strings of length 2 that each starts with a 1. Continue creating child nodes until we reach length , at which point we stop. There are a total of 2 +nodes in this tree and 2 of them are leaves (vertices of a tree with degree 1, i.e. childless nodes). Note that the parent of any node is a prex to that node. Label each node vs with the string s, where s is a binary sequence of length . See Figure 3.21 for an example when = 2.
Figure 3.21: Tree representation of the binary code B2. In general, if C is a code contained in B , then to create the tree for C, start with the tree for B. First, remove all nodes associated to a binary string for which it and
all of its descendants are not in C. Next, remove all labels which do not correspond to codewords in C. The resulting labeled graph is the tree associated to the binary code C. For visualizing the construction of Human codes later, it is important to see that we can reverse this construction to start from such a binary tree and recover a binary code from it. The codewords are determined by the following rules:
The root node gets the empty codeword. Each left-ward branch gets a 0 appended to the end of its parent. Each right-ward branch gets a 1 appended to the end.

Uniquely decodable codes

If c : A B is a code, then we can extend c to A by concatenation: c(a1 a2 ak ) = c(a1 )c(a2 ) c(ak ). If the extension c : A T is also an injection, then c is called uniquely decodable. Example 3.20. Is the Morse code in Table 3.1 uniquely decodable? Why or why not? Solution. Note that these Morse codewords all have lengths less than or equal to 4. Other commonly occurring symbols used (the digits 0 through 9, punctuation symbols, and some others) are also encodable in Morse code, but they use longer codewords. Let A denote the English alphabet, B = {0, 1} the binary alphabet, and c : A B the Morse code. Since c(ET ) = 01 = c(A), it is clear that the Morse code is not uniquely decodable. In fact, prex-free implies uniquely decodable. Theorem 3.21. If a code c : A B is prex-free, then it is uniquely decodable. Proof. We use induction on the length of a message. We want to show that if x1 xk and y1 y are messages with c(x1 ) c(xk ) = c(y1 ) c(y ), then x1 xk = y1 y. This in turn implies k = and xi = yi for all i. The case of length 1 follows from the fact that c : A B is injective (by the denition of code). Suppose that the statement of the theorem holds for all codes of length < m. We must show that the length m case is true. Suppose c(x1 ) c(xk ) = c(y1 ) c(y ), where m = max(k, ). These strings are equal, so the substring c(x1 ) of the left-hand side and the substring c(y1 ) of the right-hand side are either equal or one is contained in the other. If, say, c(x1 ) is properly contained in c(y1 ), then c is not prex-free. Likewise if c(y1 ) is properly contained in c(x1 ). Therefore, c(x1 ) = c(y1 ), which implies x1 = y1. Now remove this codeword from both sides, so c(x2 ) c(xk ) = c(y2 ) c(y ). By the induction hypothesis, x2 xk = y2 y. These facts together imply k = and xi = yi for all i. Consider now a weighted alphabet (A, p), where p : A [0, 1] satises aA p(a) = 1, and a code c : A B . In other words, p is a probability distribution on A. Think

In probability terminology, this is the expected value E(X) of the random variable X, which assigns to a randomly selected symbol in A the length of the associated codeword in c.
If there are no remaining symbols in A, label the parent a with the empty set and stop. Otherwise, go to the rst step.
These ideas are captured in Algorithm 3.6, which outlines steps to construct a binary tree corresponding to the Human code of an alphabet. Line 2 initializes a minimumpriority queue Q with the symbols in the alphabet A. Line 3 creates an empty binary tree that will be used to represent the Human code corresponding to A. The for loop from lines 4 to 10 repeatedly extracts from Q two elements a and b of minimum weights. We then create a new vertex z for the tree T and also let a and b be vertices of T. The weight W [z] of z is the sum of the weights of a and b. We let z be the parent of a and b, and insert the new edges za and zb into T. The newly created vertex z is now inserted into Q with priority W [z]. After n 1 rounds of the for loop, the priority queue has only one element in it, namely the root r of the binary tree T. We extract r from Q (line 11) and return it together with T (line 12). Algorithm 3.6: Binary tree representation of Human codes. Input : An alphabet A of n symbols. A weight list W of size n such that W [i] is the weight of ai A. Output: A binary tree T representing the Human code of A and the root r of T.
n |A| QA /* minimum priority queue */ T empty tree for i 1, 2,. , n 1 do a extractMin(Q) b extractMin(Q) z node with left child a and right child b add the edges za and zb to T W [z] W [a] + W [b] insert z into priority queue Q r extractMin(Q) return (T, r)
The runtime analysis of Algorithm 3.6 depends on the implementation of the priority queue Q. Suppose Q is a simple unsorted list. The initialization on line 2 requires O(n) time. The for loop from line 4 to 10 is executed exactly n 1 times. Searching Q to determine the element of minimum weight requires time at most O(n). Determining two elements of minimum weights requires time O(2n). The for loop requires time O(2n2 ), which is also the time requirement for the algorithm. An ecient implementation of the priority queue Q, e.g. as a binary minimum heap, can lower the running time of Algorithm 3.6 down to O(n log2 (n)). Algorithm 3.6 represents the Human code of an alphabet as a binary tree T rooted at r. To determine the actual encoding of each symbol in the alphabet, we feed T and r to Algorithm 3.7 to obtain the encoding of each symbol. Starting from the root r whose designated label is the empty string , the algorithm traverses the vertices of T in a breadth-rst search fashion. If v is an internal vertex with label e, the label of its left-child is the concatenation e0 and for the right-child of v we assign the label e1. If v happens to be a leaf vertex, we take its label to be its Human encoding. Any Human encoding assigned to a symbol of an alphabet is not unique. Either of the two children of

Algorithm 3.11: Bottom-up traversal. Input : An ordered tree T on n > 0 vertices. Output: A list of the vertices of T in bottom-up order.
Q empty queue r root of T C [0, 0,. , 0] /* n copies of 0 */ for each edge (u, v) E(T ) do C[u] C[u] + 1 R empty queue enqueue(R, r) while length(R) > 0 do v dequeue(R) for each w children(v) do if C[w] = 0 then enqueue(Q, w) else enqueue(R, w) L [] while length(Q) > 0 do v dequeue(Q) append(L, v) if v = r then C[parent(v)] C[parent(v)] 1 if C[parent(v)] = 0 then u parent(v) enqueue(Q, u) return L
Algorithm 3.12: In-order traversal. Input : A binary tree T on n > 0 vertices. Output: A list of the vertices of T in in-order.
L [] S empty stack r root of T push(S, r) while length(S) > 0 do v pop(S) if v has a left-child then u left-child of v push(S, u) append(L, v) if v has a right-child then u right-child of v push(S, u) return L

3.7. Problems

traversal: start at the root, then traverse the left and right subtrees in in-order. For this reason, in-order traversal is sometimes referred to as symmetric traversal. Our discussion is summarized in Algorithm 3.12. Since each vertex is pushed and popped exactly once, it follows that in-order traversal runs in time O(n). Using Algorithm 3.12, an in-order traversal of the tree in Figure 3.22(b) is , 1, 11, 10, 0, 01, 00, 001, 000, 0001, 0000.
3.1. Construct all nonisomorphic trees of order 7. 3.2. Let G be a weighted connected graph and let T be a subgraph of G. Then T is a maximum spanning tree of G provided that the following conditions are satised: (a) T is a spanning tree of G. (b) The total weight of T is maximum among all spanning trees of G. Modify Kruskals, Prims, and Borvkas algorithms to return a maximum spanning u tree of G. 3.3. Describe and present pseudocode of an algorithm to construct all spanning trees of a connected graph. What is the worst-case runtime of your algorithm? How many of the constructed spanning trees are nonisomorphic to each other? Repeat the exercise for minimum and maximum spanning trees. 3.4. The solution of Example 3.3 relied on the following result: Let T = (V, E) be a tree rooted at v0 and suppose v0 has exactly two children. If maxvV deg(v) = 3 and v0 is the only vertex with degree 2, then T is a binary tree. Prove this statement. Give examples of graphs that are binary trees but do not satisfy the conditions of the result. Under which conditions would the above test return an incorrect answer? 3.5. What is the worst-case runtime of Algorithm 3.1? 3.6. Figure 3.4 shows two nonisomorphic spanning trees of the grid graph. (a) For each n = 1, 2,. , 7, construct all nonisomorphic spanning trees of the n n grid graph. (b) Explain and provide pseudocode of an algorithm for constructing all spanning trees of the n n grid graph, where n > 0. (c) In general, if n is a positive integer, how many nonisomorphic spanning trees are there in the n n grid graph? (d) Describe and provide pseudocode of an algorithm to generate a random spanning tree of the n n grid graph. What is the worst-case runtime of your algorithm?

Initialize f (u, v) = 0, for all edges (u, v) E While there is a path p from s to t in Gf , such that cf (e) > 0, for all edges e E: Find cf (p) = min{cf (u, v) | (u, v) p}, For each edge (u, v) : f (u, v) = f (u, v) + cf (p) f (v, u) = f (v, u) cf (p) To prove the max-ow/min-cut theorem we will use the following lemma.
Lemma 5.21. Let G = (V, E) be a directed graph with edge capacity c : E Z, a source s V , and a sink t V. A ow f : E Z is a maximum ow if and only if there is no f -augmenting path in the graph.
In other words, a ow f in a capacitated network is a maximum ow if and only if there is no f -augmenting path in the network. Solution. One direction is easy. Suppose that the ow is a maximum. If there is an f -augmenting path then the current ow can be increased using that path, so the ow would not be a maximum. This contradiction proves the only if direction. Now, suppose there is no f -augmenting path in the network. Let S be the set of vertices v such that there is an f -unsaturated path from the source s to v. We know s S and (by hypothesis) t S. Thus there is a cut of the form (S, T ) in the network. / Let e = (v, w) be any edge in this cut, v S and w T. Since there is no f -unsaturated path from s to w, e is f -saturated. Likewise, any edge in the cut (T, S) is f -zero. Therefore, the current ow value is equal to the capacity of the cut (S, T ). Therefore, the current ow is a maximum. We can now prove the max-ow/min-cut theorem. Solution. Let f be a maximum ow. If S = {v V | there exists an f saturated path from s to v}, then by the previous lemma, S = V. Since T = V S is non-empty, there is a cut C = (S, T ). Each edge of this cut C in the capacitated network G is f -saturated. Here is some Python code1 which implements this. The class FlowNetwork is basically a Sage Graph class with edge weights and an extra data structure representing the ow on the graph.
class Edge : def __init__ ( self ,U ,V , w ): self. source = U self. to = V self. capacity = w def __repr__ ( self ): return str ( self. source ) + " ->" + str ( self. to ) + " : " + str ( self. capacity ) class FlowNetwork ( object ): """ This is a graph structure with edge capacities. EXAMPLES : g = FlowNetwork () map ( g. add_vertex , [ s ,o ,p ,q ,r ,t ]) g. add_edge ( s ,o ,3) g. add_edge ( s ,p ,3) g. add_edge ( o ,p ,2) g. add_edge ( o ,q ,3) g. add_edge ( p ,r ,2) g. add_edge ( r ,t ,3) g. add_edge ( q ,r ,4) g. add_edge ( q ,t ,2) print g. max_flow ( s ,t ) """ def __init__ ( self ): self. adj , self. flow , = {} ,{} def add_vertex ( self , vertex ): self. adj [ vertex ] = [] def get_edges ( self , v ): return self. adj [ v ]
Please see http://en.wikipedia.org/wiki/Ford-Fulkerson_algorithm.

5.4. Mengers Theorem

def add_edge ( self , u ,v , w =0): assert ( u != v ) edge = Edge (u ,v , w ) redge = Edge (v ,u ,0) edge. redge = redge redge. redge = edge self. adj [ u ]. append ( edge ) self. adj [ v ]. append ( redge ) self. flow [ edge ] = self. flow [ redge ] = 0

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To use this License in a document you have written, include a copy of the License in the document and put the following copyright and license notices just after the title page: Copyright YEAR YOUR NAME. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no BackCover Texts. A copy of the license is included in the section entitled GNU Free Documentation License. If you have Invariant Sections, Front-Cover Texts and Back-Cover Texts, replace the with. Texts. line with this: with the Invariant Sections being LIST THEIR TITLES, with the FrontCover Texts being LIST, and with the Back-Cover Texts being LIST. If you have Invariant Sections without Cover Texts, or some other combination of the three, merge those two alternatives to suit the situation. If your document contains nontrivial examples of program code, we recommend releasing these examples in parallel under your choice of free software license, such as the GNU General Public License, to permit their use in free software.

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Cn , 15 En , 41 Gc , 33 Kn , 14 Km,n , 16 Ln , 34 Pn , 15, 34 Qn , 34 Wn , 28 , 28 (G), 9 adj, 4 L, =, deg, 5, 9 deg+ , 6 deg , 6 (G), 9, 160 depth(v), 93 dir, 82 height(T ), 93 iadj, 5 id, 5 (G), (G), (G), 159 (G), 159 lg, 141 oadj, 5 od, 5 , 13 G, 33 , 34 , 122, 125 (n), 102 cut(S, T ), 160 f -augmenting, 161 f -saturated, 161 f -unsaturated, 161 f -zero, 161 n-space, 119 graph6, 45, 47, 49, 50 sparse6, 47, 49 Lukaszewicz, J., 110 acyclic, 93, 104, 105, 132 graph random, 177 adjacency matrix, 18 reduced, 19 algorithm greedy, 67, 68, 104, 107 optimization, 98 random, 43, 92, 131134, 177 recursive, 137 alphabet, 117, 121 binary, 122, 123 English, 117, 134 weighted, 117, 124 arcs, 3 Argentina, 84, 85 articulation point, 157 ASCII, 47, 49, 117 augmenting path, 161 Australia, 84, 85 Australian National University, 47 automata theory, 75 backtrack, 55 balanced bracket problem, 88, 89 Bangkok, 84, 85 Baudot, E., 118 Beijing, 84, 85 Bellman, Richard E., 68 Bellman-Ford algorithm, 63, 6871, 75, 77, 83 Berlin, 84, 85 BFS, 5055, 57, 61, 62 big-endian, 47, 49, 50 Biggs, Norman, 124 binary search tree, 150 binary tree, 95, 97, 114116, 134 206
Index complete, 114, 141 level, 141 random, 116, 134 bipartite graph, 15, 43 complete, 16 bit, 47, 117, 122 least signicant, 47 most signicant, 47 parity, 47 bit vector, 47, 49, 50 length, 47 block, 157 bond, 31, 98 Bondys theorem, 158 Borvka u algorithm, 104, 110, 112, 131, 132, 137 Otakar, 104, 110 bowtie graph, 12 braille, 117 branch cut, 95, 97 Brasilia, 84, 85 Brazil, 84, 85 breadth-rst search, 5055, 60, 61, 63, 64, 82, 97, 125, 127 tree, 51, 54 bridge, 31, 93, 99, 110 bridgeless, 158 BST, 150 Buenos Aires, 84, 85 buttery graph, 12 Canada, 84, 85 canonical label, 23, 24 capacity, 161 cut, 162 card, 57 cardinality, 9 Carroll, Lewis, 2 Cartesian product, 34 Catalan number, 41, 42, 115 recursion, 115 check matrix, 19 chemistry, 74 chess, 55, 117 chessboard, 55 knight, 55 knight piece, 55 knights tour, 55, 56, 90 child left, 114, 125, 129 right, 114, 125, 129 China, 84, 85 Chinese ring puzzle, 118, 119 Choquet, G., 110 Chvtal graph, 132, 133 a circuit, 12 board, 50 electronic, 103 classication tree, 94, 98 closed form, 115 code, 117, 122 binary, 117, 122, 123 block, 117 economy, 117 error-correcting, 19, 117 linear, 119 optimal, 124 prex, 117 prex-free, 117, 123, 134 reliability, 117 security, 117 tree representation, 122 uniquely decodable, 123 variable-length, 117 codeword, 117, 122 length, 124 coding function, 117 Cohen, Danny, 49 Collatz conjecture, 134, 135 graph, 135, 136 length, 135 sequence, 135 tree, 135, 136 color code, 117 coloring edge, 37 vertex, 37, 38 combinatorial graphs, 2 combinatorics, 119 complement, 33 complete graph, 14, 133, 134 component, 13, 27, 99 connected, 104 computer science, 126