Contents of Chapter 7 Chapter 7 Backtracking –7.1 The General method –7.2 The 8-queens problem –7.3 Sum of subsets –7.4 Graph coloring –7.5 Hamiltonian.

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Contents of Chapter 7

• Chapter 7 Backtracking– 7.1 The General method– 7.2 The 8-queens problem– 7.3 Sum of subsets– 7.4 Graph coloring– 7.5 Hamiltonian cycles– 7.6 Knapsack problem– 7.7 References and readings– 7.8 Additional exercises

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7.1 The General Method

• Backtracking – One of the most general techniques for searching a set of

solutions or for searching an optimal solution– The name first coined by D.H. Lehmer

• Basic idea of backtracking– Desired solution expressed as an n-tuple (x1,x2,…,xn) where xi

are chosen from some set Si

• If |Si|=mi, m=m1m2..mn candidates are possible

– Brute force approach• Forming all candidates, evaluate each one, and saving the

optimum one– Backtracking

• Yielding the same answer with far fewer than m trials• “Its basic idea is to build up the solution vector one

component at a time and to use modified criterion function Pi(x1,..,xi) (sometimes called bounding function) to test whether the vector being formed has any chance of success. The major advantage is: if it is realized that the partial vector (x1,..,xi) can in no way lead to an optimum solution, then mi+1…mn possible test vectors can be ignored entirely.”

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• Constraints – Solutions must satisfy a set of constraints– Explicit vs. implicit– Definition 1: Explicit constraints are rules that restrict

each xi to take on values only from a given set.

• eg)xi 0 or Si = { all nonnegative real

numbers }xi = 0 or 1 or Si = { 0, 1 }

li xi ui or Si = {a : li a ui }

• All tuples satisfying the explicit constraints define a

possible solution space for I (I=problem instance)– Definition 2: The implicit constraints are rules that

determine which of the tuples in the solution space of I satisfy the criterion function. Thus implicit constrains describe the way in which the xi must relate to each other.

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• Example 7.1 (8-queens) – Classic combinatorial problem

• Place eight queens on an 8*8 chessboard so that no two attack

– Without loss of generality, assume that queen i is placed on row i

– All solutions represented as 8-tuples (x1,x2,…,x8) where xi is the column on which queen i is placed

– Constraints• Explicit constraints

– Si={1,2,…,8}

– So, solution space consists of 88 8-tuples• Implicit constraints

– No two xi’s can be the same (By this, solution space reduced from 88 to 8!)

– No two queens can be on the same diagonal

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• Example 7.1 (Continued) – Figure 7.1

• 8-tuple = (4, 6, 8, 2, 7, 1, 3, 5)

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• Example 7.2 (sum of subsets) – Given positive numbers wi, 1≤i≤n, and m, find all

subsets of the wi whose sum is m

• eg) (w1,w2,w3,w4) = (11,13,24,7) and m=31

– Solutions are (11,13,7) and (24,7)– Representation of solution vector

• Variable-sized– By giving indices– In the above example, (1,2,4) and (3,4)

– Explicit constraints: xi ∈ { j | j is integer and 1≤j≤n}

– Implicit constraints: no two be the same, the sums are m

• Fixed-sized

– n-tuple (x1,x2,…,xn) where xi∈{0,1}, 1≤i≤n

– In the above example, (1,1,0,1) and (0,0,1,1)

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• Backtracking– Systematically searching the solution space by using a

tree organization

• Example 7.3 (n-queens)– Example of 4-queens problem (Figure 7.2)

• 4!=24 leaf nodes

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• Example 7.4 (sum of subsets)– Variable tuple size (Figure 7.3)

• Solution space defined by all paths from the root to any node

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• Example 7.4 (Continued)– Fixed tuple size (Figure 7.4)

• Solution space defined by all paths from the root to a leaf node

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• Terminology– The tree organization of the solution space is state space

tree– Each node in the state space tree defines problem state– Solution state are those problem states s for which the

path from the root to s defines a tuple in the solution space • eg)

– In Figure 7.3, all nodes are solution state– In Figure 7.4, only leaf nodes are solution state

– Answer state are those solution states s for which the path from root to s defines a tuple that is a member of solutions (it satisfies the implicit constraints)

– Solution space is partitioned into disjoint sub-solution space at each internal node

– Static vs. dynamic tree• Static trees are independent of the problem instance• Dynamic trees are dependent of the problem instance

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• Terminology– A node which has been generated and all of whose

children have not yet been generated is called a live node

– The live node whose children are currently being generated is called E-node

– A dead node is a generated node which is not to be expanded further or all of whose children have been generated

• Two different ways of tree generation– Backtracking (Chapter 7)

• Depth first node generation with bounding functions– Branch-and-bound (Chapter 8)

• E-node remains E-node until it is dead

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• Example 7.5 (4-queens)– This example shows how the backtracking works– Figure6 7.5

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– Figure 7.6

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• Formulation of the backtracking– T(x1,x2,…,xi): set of all possible values for xi+1 such that

(x1,x2,…,xi,xi+1) is also a path to a problem state

– Bi+1(x1,x2,…,xi,xi+1): bounding function

• If Bi+1(x1,x2,…,xi,xi+1) is false, then the path cannot be extended to reach an answer node

– Recursive backtracking algorithm (Program 7.1)• Initially invoked by Backtrack(1)

void Backtrack( int k )// This is a schema that describes the backtracking// process using recursion. On entering, the first k-1// values x[1], x[2], …., x[k-1] of the solution vector// x[1:n] have been assigned. x[] and n are global.{ for (each x[k] such that x[k] T(x[1], …, x[k-1]) { if (Bk (x[1], x[2], …, x[k])) { if (x[1], x[2], …, x[k] is a path to an answer node) output x[1:k]; if (k < n) Backtrack(k+1); } } }

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– Iterative backtracking algorithm (Program 7.2)

void IBacktrack( int n )// This is a schema that describes the backtracking process.// All solutions are generated in x[1:n] and printed// as soon as they are determined.{ int k=1; while (k) { if (there remains an untried x[k] such that x[k] is in T( x[1], x[2], …, x[k-1] ) and Bk(x[1], …,x[k]) is true) { if ( x[1], …, x[k] is a path to an answer node ) output x[1:k]; k++; // Consider the next set. } else k--; // Backtrack to the previous set. } }

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• Four factors for efficiency of the backtracking algorithms– Time to generate the next xk

– Number of xk satisfying the explicit constraints

– Time for the bounding function Bk

– Number of xk satisfying Bk

* The first three relatively independent of the problem instance, and the last dependent

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7.2 The 8-Queens Problem

• Bounding function– No two queens placed on the same column xi’s are

distinct– No two queens placed on the same diagonal how to

test?• The same value of “row-column” or “row+column”• Supposing two queens on (i,j) and (k,l)

– i-j=k-l (i.e., j-l=i-k) or i+j=k+l (i.e., j-l=k-i)• So, two queens placed on the same diagonal iff

|j-l| = |i-k|– Bounding function: can a new queen be placed ?

(Program 7.4)

bool Place(int k, int i) // Returns true if a queen can be placed in kth row and // ith column. Otherwise it returns false. x[] is a // global array whose first (k-1) values have been set. // abs(r) returns the absolute value of r. { for (int j=1; j < k; j++) if ((x[j] == i) // Two in the same column || (abs(x[j]-i) == abs(j-k))) // or in the same diagonal return(false); return(true); }

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7.2 The 8-Queens Problem

– Backtracking algorithm for the n-queens problem (Program 7.5)• Invoked by Nqueens(1,n)

void NQueens(int k, int n) // Using backtracking, this procedure prints all // possible placements of n queens on an nXn // chessboard so that they are nonattacking. { for (int i=1; i<=n; i++) { if (Place(k, i)) { x[k] = i; if (k==n) { for (int j=1;j<=n;j++) cout << x[j] << ' '; cout << endl;} else NQueens(k+1, n); } } }

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7.3 Sum of Subsets

• Bounding function– Bk(x1,..,xk)=true iff

– Assuming wi’s in nondecreasing order, (x1,..,xk) cannot

lead to an answer node if

– So, the bounding function is

mwxwn

1kii

k

1iii

mwxw 1k

k

1iii

mwxwand

mwxwifftrue)x,...,x(B

1k

k

1iii

n

1kii

k

1iiik1k

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7.3 Sum of Subsets

• Backtracking algorithm for the sum of sunsets (Program 7.6)– Invoked by SumOfSubsets(0,1,∑i=1,nwi) void SumOfSub(float s, int k, float r)

// Find all subsets of w[1:n] that sum to m. The values // of x[j], 1<=j<k, have already been determined. // s= ∑j=1,k-1

w[j]*x[j] and r= ∑j=k,n w[j].

// The w[j]'s are in nondecreasing order. // It is assumed that w[1]<=m and ∑i=1

n w[i]>=m. { // Generate left child. Note that s+w[k] <= m // because Bk-1 is true. x[k] = 1; if (s+w[k] == m) { // Subset found for (int j=1; j<=k; j++) cout << x[j] << ' '; cout << endl; } // There is no recursive call here as w[j] > 0, 1 <= j <= n. else if (s+w[k]+w[k+1] <= m) SumOfSub(s+w[k], k+1, r-w[k]); // Generate right child and evaluate Bk-1. if ((s+r-w[k] >= m) && (s+w[k+1] <= m)) { x[k] = 0; SumOfSub(s, k+1, r-w[k]); } }

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7.3 Sum of Subsets

• Example 7.6 (Figure 7.10)– n=6, w[1:6]={5,10,12,13,15,18}, m=30

Contents of Chapter 7 Chapter 7 Backtracking –7.1 The General method –7.2 The 8-queens problem –7.3 Sum of subsets –7.4 Graph coloring –7.5 Hamiltonian.

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