A Little Bit of Classics_ Dynamic Programming Over Subsets and Paths in Graphs - Codeforces

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6/5/2014 A little bit of classics: dynamic programming over subsets and paths in graphs - Codeforces

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A little bit of classics: dynamic programmingoversubsets and paths in graphs

Author thanks adamaxfor translation this article into English.

Introduction

After Codeforces Beta Round #11several participants expressed a wish to read something

about problems similar to problem Dof that round. The author of this article, for the purpose

of helping them, tried searching for such information, but to his surprise couldn't find

anything on the Internet. It is not known whether the search was not thorough or there'sreally nothing out there, but (just in case) the author decided to write his own article on this

topic.

In some sense this article may be regarded as a tutorial for the problem Dfrom Beta Round

#11.

In this article quite well-known algorithms are considered: the search for optimal

Hamiltonian walks and cycles, finding their number, check for existence and something

more. The so-called "dynamic programming over subsets" is used. This method requires

exponential time and memory and therefore can be used only if the graph is very small -

typically 20 vertices or less.

DP over subsets

Consider a set of elements numbered from 0 to N-1. Each subset of this setcan beencoded by a sequence of N bits (we will call this sequence "a mask"). The i-th element

belongs to the subset if and only if the i-th bit of the mask equals 1. For instance, the mask

00010011means that the subset of the set [0... 7]consists of elements 0, 1 and 4. There

are totally 2 masks, and so 2 subsets. Each mask is in fact an integer number written in

binary notation.

The method is to assign a value to each mask (and, therefore, to each subset) and

compute the values for new masks using already computed values. As a rule, to find the

value for a subsetAwe remove an element in every possible way and use values for

obtained subsetsA' ,A' ,... ,A' to compute the value forA. This means that the values

forA'must have been computed already, so we need to establish an ordering in which

masks will be considered. It's easy to see that the natural ordering will do: go over masks in

increasing order of corresponding numbers.

We will use the following notation:

bit(i,mask)- the i-th bit of maskcount(mask)- the number of non-zero bits in mask

first(mask)- the number of the lowest non-zero bit in mask

(a?b:c)- returns bif aholds, or c otherwise.

The elements of our set will be vertices of the graph. For the sake of simplicity we'll assumethat the graph is undirected. Modification of the algorithms for directed graphs is left as an

exercise for the reader.

1. Search for the shortest Hamiltonian walk

Let the graph G=(V,E)have nvertices, and each edge have a weight

Before contestCodeforces Round #252 (Div. 2)

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d(i,j). We want to find a Hamiltonian walk for which the sum of weights of its edges isminimal.

Let dp[mask][i]be the length of the shortest Hamiltonian walk in the subgraph generated

by vertices in mask, that ends in the vertex i.

The DP can be calculated by the following formulas:

dp[mask][i]=0, if count(mask)=1and bit(i,mask)=1, if

count(mask)>1and bit(i,mask)=1dp[mask][i]=in other cases.

Now the desired minimal length is . Ifp =,then there is no Hamiltonian walk in the graph. Otherwise it's easy to recover the walk itself.

Let the minimal walk end in the vertex i. Then the vertexji, for which, is the previous vertex in the path.

Now remove ifrom the set and find the vertex previous tojin the same way. Continuing

this process until only one vertex is left, we'll find the whole Hamiltonian walk.

This solution requires O(2 n)of memory and O(2 n )of time.

2. Finding the number of Hamiltonian walks

Let the graph G=(V,E)be unweighted. We'll modify the previous algorithm. Let

dp[mask][i]be the number of Hamiltonian walks on the subset mask, which end in the

vertex i. The DP is rewritten in the following way:

dp[mask][i]=1, if count(mask)=1and bit(i,mask)=1

, if count(mask)>1and

bit(i,mask)=1dp[mask][i]=0in other cases.

The answer is .


3. Finding the number of simple paths

Calculate the DP from the previous paragraph. The answer is

. The coefficient 1/2is required because

each simple path is considered twice - in both directions. Also note that only paths of

positive length are taken into account. You can add nzero-length paths, of course.


4. Check for existence of Hamiltonian walk

We can use solution 2 replacing the sum with bitwise OR. dp[mask][i]will contain a

boolean value - whether there exists a Hamiltonian walk over the subset maskwhich endsin the vertex i. DP is the following:


, if count(mask)>1and

bit(i,mask)=1

dp[mask][i]=0in other cases.

This solution, like solution 2, requires O(2 n)of memory and O(2 n )of time. It can be

improved in the following way.

Let dp'[mask]be the mask of the subset consisting of those vertices jfor which there

exists a Hamiltonian walk over the subset maskending inj. In other words, we 'compress'

the previous DP: dp'[mask]equals . For the graph Gwriteout n masksM, which give the subset of vertices incident to the vertex i. That is,

.

DP will be rewritten in the following way:

dp'[mask]=2, if count(mask)=1and bit(i,mask)=1

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, if

count(mask)>1

dp'[mask]=0in other cases.

Pay special attention to the expression . The first part of the

expression is the subset of vertices j, for which there exists a Hamiltonian walk over the

subset mask minus vertex i, ending inj. The second part of the expression is the set of

vertices incident to i. If these subsets have non-empty intersection (their bitwise AND is

non-zero), then it's easy to see that there exists a Hamiltonian walk in maskending in the

vertex i.

The final test is to compare dp[2 -1]to 0.

This solution uses O(2 )of memory and O(2 n)of time.

5. Finding the shortest Hamiltonian cycle

Since we don't care at which vertex the cycle starts, assume that it starts at 0. Now use

solution 1 for the subset of vertices, changing the formulas in the following way:

dp[1][0]=0

, if i>0,

bit(0,mask)=1and bit(i,mask)=1

dp[mask][i]=in other cases.

So dp[mask][i]contains the length of the shortest Hamiltonian walk over the subset mask,

starting at 0 and ending at i.

The required minimum is calculated by the formula

. If it equals , there is no Hamiltonian

cycle. Otherwise the Hamiltonian cycle can be restored by a method similar to solution 1.

6. Finding the number of Hamiltonian cycles

Using ideas from solutions 5 and 2 one can derive a DP calculating the number of

Hamiltonian cycles requiring O(2 n )of time and O(2 n)of memory.

7. Finding the number of simple cycles

Let dp[mask][i]be the number of Hamiltonian walks over the subset mask, starting at the

vertex first(mask)and ending at the vertex i. DP looks like this:


, if count(mask)>1,

bit(i,mask)=1and ifirst(mask)dp[mask][i]=0 otherwise.

The answer is .

This solution requres O(2 n )of time and O(2 n)of memory.

8. Checking for existence of Hamiltonian cycle

We can modify solution 5 and, using the trick from solution 4, obtain an algorithm requiring

O(2 n)of time and O(2 )of memory.

Exercises

CFBR11D

CCTOOLS

P.S.This article may be extended and fixed in future. The author would be grateful for

supplementing the Exercises section and for pointing out any mistakes and inaccuracies.

algorithms, dynamic programming, graphs

n

n n

n 2 n

n 2 n

n n

+22 Ripatti 4 years ago 13
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Comments (13)

AlexErofeev

Ripatti

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Write comment?

4 years ago, # | +3http://www.spoj.pl/problems/ASSIGN/

For example

Reply

4 years ago, # ^ | 0, )) ,

, .

, .

. , , .

:)

Reply

4 years ago, # | 0Nice tutorial. You could also add an explanation on how to count

hamiltonian circuits on a grid of size n x m with (n


5/5



Codeforces(c) Copyright 2010-2014 Mike MirzayanovThe only programming contests Web 2.0 platform

Server time: 2014-06-05 11:42:28 (c4).

havaliza

Ripatti

saadtaame

bufferedreader

niklasb

. .

is an example problem for part 5.

Reply

4 years ago, # ^ | 0Thanx! Very nice problem)) I added it.

Reply

new, 3 days ago, # | 0

Could you please further explain the recurrence in the first example (finding

a shortest Hamiltonian walk)?

Reply

new, 3 days ago, # ^ | 0

For me too, please.Reply

new, 3 days ago, # | Rev. 3 0

The following paper is potentially useful as reference:

"A Dynamic Programming Approach to Sequencing Problems" by Michael

Held and Richard M. Karp, 1962. [citation, download]

It contains lots of preliminary analysis and at least the DP approaches

described in 1. and 5. of your post. I believe the algorithm is actually called

"HeldKarp algorithm" in academic literature, but it was described

independently by Bellman et al. in another paper.

A list of optimal algorithms for TSP with better runtime complexity can be

found here.

Reply
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A Little Bit of Classics_ Dynamic Programming Over Subsets and Paths in Graphs - Codeforces