WHY TRANSITIVE CLOSURE IS IMPORTANT?

1. WHYTRANSITIVE CLOSUREIS IMPORTANT? APPLICATION IN PAGING NETWORKS + n NODES IN A PAGING NETWORK + WHO CAN REACH WHO – INDICATED BY ARCS + A PATH SHOWS WHO CAN BE REACHED BY A CHAIN OF ACTIONS ( NOT NECESSARILY DIRECT) TRANSITIVE CLOSURE SHOWS DIRECTLY WHO CAN REACH WHO ( DIRECTLY OR INDIRECTLY ) HOW TO COMPUTE TRANSITIVE CLOSURE? 1. DEPTH FIRST SEARCH ( Inefficient ) O ( nm ) 2. USING SHORT CUTS ( Better ) O ( n3 ) 3. WARSHALL ALGORITHM ( Best ) O ( n3 / c )

2. CONSTRUCTING TRANSITIVE CLOSURE DFS FOR TRANSITIVE CLOSURE IDEA: Build a DFS tree from each of the vertices ( n ) IN MATRIX NOTATION: Fill in row by row the new R adjacency matrix . DFS ALGORITHM FOR TC OF G D = { S, A } BUILD D* = { S, R } INPUT D OUTPUT R, Initialized to zero While (i < n) Fill in Row R[i] with the nodes discovered by DFS at this step Return R Efficiency Repeats too many steps and does not make use of discovered dependencies at a previous row search. N iterations, O(m) operations at each.

3. CONSTRUCTING TRANSITIVE CLOSURE SHORTCUTS FOR TC Say (of length 2) - Example 1 2 3 4 5 6 7 8 If there is edge Sij and also Sjk then we insert Sik BOOLEAN REPRESENTATIONS OF SHORT CUTS We update elements of matrix R rij = rij (rik rkj )  If both rik and rkj are TRUE, then this is TRUE and write 1 into rij otherwise leave rij unchanged. Efficiency : O ( nm )

4. BOOLEAN ALGEBRA OPERATIONS TRUTH TABLES: OR AT LEAST ONE IS TRUE AND BOTH ARE TRUE

5. EXAMPLE: X ( Y Z) Y X Z CIRCUIT DIAGRAM

6. CONSTRUCTING TRANSITIVE CLOSURE Transitive Closure by shortcuts: Input: A and n, where A is an n X n boolean matrix that represents a binary relation. Output: R, the boolean matrix for the transitive closure of A void simpleTransitiveClosure ( boolean[][] A, int n, boolean[][] R) int i,j,k; Copy A into R Set all main diagonal entries, rii, to TRUE While ( any entry of R changed during one complete pass ) for( i=1; i<=n; i++) for( j=1; j<=n; j++) for( k=1; k<=n; k++) rij = rij (rik rkj )

7. CONSTRUCTING TRANSITIVE CLOSURE WARSHALL’S ALGORITHM FOR TC • Uses shortcuts • Orders and calculates them in a more clever way • Puts the internal cycle outside rij = rij (rik rkj ) + We process each i  k  j triple only once (ordered by k ) + Preciously we did twice ( in shortcut ) once for the incoming and also for outgoing arcs.

8. CONSTRUCTING TRANSITIVE CLOSURE THE BASIC ALGORITHM (WARSHALL) Input: A and n, where A is an n X n matrix that represents a binary relation. Output: R, the n X n matrix for the transitive closure of A void transitiveClosure ( boolean[][] A, int n, boolean[][] R) int i,j,k; Copy A into R Set all main diagonal entries, rii, to true for( k=1; k<=n; k++) for( i=1; i<=n; i++) for( j=1; j<=n; j++) rij = rij (rik rkj ) Complexity : Still O( n3), but with smaller constant O( n3 / c ), where c is the word size in the boolean operation.

9. Bit String, Bit Matrix, bitwiseOR A bit String of length n is a sequence of n bits occupying contiguous storage, beginning on a computer word boundary, and being padded out to a computer-word boundary at the end, if needed, That is, if a computer holds c bits, then a bit string of n bits is stored in an array of [n/c] computer words. A bit matrix is an array of bit strings, with each bit string representing one row of the matrix. If A is a bit matrix, then A[i] denotes the ith row of A and is a bit string. Also, aij denotes the jth bit of A[i]. The procedure bitwiseOR(a, b, n), where a nad b are bit strings and n is an integer, is defined to compute a V b bitwise for n bits and leave the result in a.

10. Warshall Algorithm using bit matrices Input:A and n, where A is an n X n bit matrix that represents a binary relation. We assume for the pseudocode that the class BitMetrix has been defined. Output:R, the n X n matrix for the transitive closure of A, also a bit Matrix. Algorithm: void warshallBitMatrices ( bitMatrix A, int n, bitMatrix R). int i,k; Copy A into R; Set all main diagonal entries, rii, to 1; for( k=1; k<=n; k++). for( i=1; i<=n; i++). if (rik == 1). bitwiseOR(R[I], R[k], n);