![Algorithm 5.2.2 Merge This algorithm receives as input indexes i , m , and j , and an array a , where a [ i ], ... , a [ m ] and a [ m +1], ... , a [ j ] are each sorted in nondecreasing order. These two nondecreasing subarrays are merged into a single nondecreasing array.](https://image.slidesharecdn.com/chap05alg-100316020509-phpapp02/85/Chap05alg-4-320.jpg)
![Input Parameters: a , i , m , j Output Parameter: a merge ( a , i , m , j ) { p = i // index in a [ i ], ... , a [ m ] q = m + 1 // index in a [ m + 1], ... , a [ j ] r = i // index in a local array c while ( p ≤ m && q ≤ j ) { // copy smaller value to c if ( a [ p ] ≤ a [ q ]) { c [ r ] = a [ p ] p = p + 1 } else { c [ r ] = a [ q ] q = q + 1 } r = r + 1 } ...](https://image.slidesharecdn.com/chap05alg-100316020509-phpapp02/85/Chap05alg-5-320.jpg)
![... // copy remainder, if any, of first subarray to c while ( p ≤ m ) { c [ r ] = a [ p ] p = p + 1 r = r + 1 } // copy remainder, if any, of second subarray to c while ( q ≤ j ) { c [ r ] = a [ q ] q = q + 1 r = r + 1 } // copy c back to a for r = i to j a [ r ] = c [ r ] }](https://image.slidesharecdn.com/chap05alg-100316020509-phpapp02/85/Chap05alg-6-320.jpg)
![Algorithm 5.2.3 Mergesort This algorithm sorts the array a [ i ], ... , a [ j ] in nondecreasing order. It uses the merge algorithm (Algorithm 5.2.2). Input Parameters: a , i , j Output Parameter: a mergesort ( a , i , j ) { // if only one element, just return if ( i == j ) return // divide a into two nearly equal parts m = ( i + j )/2 // sort each half mergesort ( a , i , m ) mergesort ( a , m + 1, j ) // merge the two sorted halves merge ( a , i , m , j ) }](https://image.slidesharecdn.com/chap05alg-100316020509-phpapp02/85/Chap05alg-7-320.jpg)
![Algorithm 5.3.2 Finding the Distance Between a Closest Pair of Points This algorithm finds the distance between a closest pair of points. The input is an array p [1], ... , p [ n ] of n = 2 points. If p is a point, p . x is the x - coordinate of p , and p . y is the y -coordinate of p . The function merge is Algorithm 5.2.2 and mergesort is Algorithm 5.2.3. The function merge uses as the key the y -coordinate of the point. The function mergesort uses as the key either the x - or y -coordinate of the point; the comments indicate which. The function dist ( p , q ) returns the Euclidean distance between points p and q .](https://image.slidesharecdn.com/chap05alg-100316020509-phpapp02/85/Chap05alg-8-320.jpg)
![Input Parameters: p Output Parameter: None closest_pair ( p ) { n = p . last mergesort ( p ,1, n ) // sort by x-coordinate return rec_cl_pair ( p ,1, n ) } // rec_cl_pair assumes that input is sorted by x -coordinate. // At termination, the input is sorted by y -coordinate. rec_cl_pair ( p , i , j ) { if ( j - i < 3) { mergesort ( p , i , j ) // sort by y -coordinate // find the distance delta between a closest pair delta = dist ( p [ i ], p [ i + 1]) if ( j - i == 1) // two points return delta // three points if ( dist ( p [ i + 1], p [ i + 2]) < delta ) delta = dist ( p [ i + 1], p [ i + 2]) if ( dist ( p [ i ], p [ i + 2]) < delta ) delta = dist ( p [ i ], p [ i + 2]) return delta } ...](https://image.slidesharecdn.com/chap05alg-100316020509-phpapp02/85/Chap05alg-9-320.jpg)
![... k = ( i + j )/ 2 l = p [ k ]. x deltaL = rec_cl_pair ( p , i , k ) deltaR = rec_cl_pair ( p , k + 1, j ) delta = min ( deltaL , deltaR ) // p [ i ], ... , p [ k ] is now sorted by y -coordinate, and // p [ k + 1], ... , p [ j ] is now sorted by y -coordinate. merge ( p , i , k , j ) // p [ i ], ... , p [ j ] is now sorted by y -coordinate. // store points in the vertical strip in v . t = 0 for k = i to j if ( p [ k ]. x > l - delta && p [ k ]. x < l + delta ) { t = t + 1 v [ t ] = p [ k ] } // look for closer pairs in the strip by comparing // each point in the strip to the next 7 points. for k = 1 to t - 1 for s = k + 1 to min ( t , k + 7) delta = min ( delta , dist ( v [ k ], v [ s ])) return delta }](https://image.slidesharecdn.com/chap05alg-100316020509-phpapp02/85/Chap05alg-10-320.jpg)
![Algorithm 5.4.1 Matrix Product This algorithm computes the product C of the n × n matrices A and B directly from the definition of the matrix product. Input Parameters: A , B Output Parameter: C matrix_product ( A , B , C ) { n = A . last for i = 1 to n for j = 1 to n { C [ i ][ j ] = 0 for k = 1 to n C [ i ][ j ] = C [ i ][ j ] + A [ i ][ k ] * B [ k ][ j ] } }](https://image.slidesharecdn.com/chap05alg-100316020509-phpapp02/85/Chap05alg-11-320.jpg)
Chap05alg
- 1. CHAPTER 5 Divide and Conquer
- 2. Algorithm 5.1.4 Tiling a Deficient Board with Trominoes This algorithm constructs a tiling by trominoes of a deficient n × n board where n is a power of 2.
- 3. Input Parameters: n , a power of 2 (the board size); the location L of the missing square Output Parameters: None tile ( n , L ) { if ( n == 2) { // the board is a right tromino T tile with T return } divide the board into four n /2 × n /2 subboards place one tromino as in Figure 5.1.4(b) // each of the 1 × 1 squares in this tromino // is considered as missing let m 1 , m 2 , m 3 , m 4 be the locations of the missing squares tile( n /2, m 1 ) tile( n /2, m 2 ) tile( n /2, m 3 ) tile( n /2, m 4 ) }
- 4. Algorithm 5.2.2 Merge This algorithm receives as input indexes i , m , and j , and an array a , where a [ i ], ... , a [ m ] and a [ m +1], ... , a [ j ] are each sorted in nondecreasing order. These two nondecreasing subarrays are merged into a single nondecreasing array.
- 5. Input Parameters: a , i , m , j Output Parameter: a merge ( a , i , m , j ) { p = i // index in a [ i ], ... , a [ m ] q = m + 1 // index in a [ m + 1], ... , a [ j ] r = i // index in a local array c while ( p ≤ m && q ≤ j ) { // copy smaller value to c if ( a [ p ] ≤ a [ q ]) { c [ r ] = a [ p ] p = p + 1 } else { c [ r ] = a [ q ] q = q + 1 } r = r + 1 } ...
- 6. ... // copy remainder, if any, of first subarray to c while ( p ≤ m ) { c [ r ] = a [ p ] p = p + 1 r = r + 1 } // copy remainder, if any, of second subarray to c while ( q ≤ j ) { c [ r ] = a [ q ] q = q + 1 r = r + 1 } // copy c back to a for r = i to j a [ r ] = c [ r ] }
- 7. Algorithm 5.2.3 Mergesort This algorithm sorts the array a [ i ], ... , a [ j ] in nondecreasing order. It uses the merge algorithm (Algorithm 5.2.2). Input Parameters: a , i , j Output Parameter: a mergesort ( a , i , j ) { // if only one element, just return if ( i == j ) return // divide a into two nearly equal parts m = ( i + j )/2 // sort each half mergesort ( a , i , m ) mergesort ( a , m + 1, j ) // merge the two sorted halves merge ( a , i , m , j ) }
- 8. Algorithm 5.3.2 Finding the Distance Between a Closest Pair of Points This algorithm finds the distance between a closest pair of points. The input is an array p [1], ... , p [ n ] of n = 2 points. If p is a point, p . x is the x - coordinate of p , and p . y is the y -coordinate of p . The function merge is Algorithm 5.2.2 and mergesort is Algorithm 5.2.3. The function merge uses as the key the y -coordinate of the point. The function mergesort uses as the key either the x - or y -coordinate of the point; the comments indicate which. The function dist ( p , q ) returns the Euclidean distance between points p and q .
- 9. Input Parameters: p Output Parameter: None closest_pair ( p ) { n = p . last mergesort ( p ,1, n ) // sort by x-coordinate return rec_cl_pair ( p ,1, n ) } // rec_cl_pair assumes that input is sorted by x -coordinate. // At termination, the input is sorted by y -coordinate. rec_cl_pair ( p , i , j ) { if ( j - i < 3) { mergesort ( p , i , j ) // sort by y -coordinate // find the distance delta between a closest pair delta = dist ( p [ i ], p [ i + 1]) if ( j - i == 1) // two points return delta // three points if ( dist ( p [ i + 1], p [ i + 2]) < delta ) delta = dist ( p [ i + 1], p [ i + 2]) if ( dist ( p [ i ], p [ i + 2]) < delta ) delta = dist ( p [ i ], p [ i + 2]) return delta } ...
- 10. ... k = ( i + j )/ 2 l = p [ k ]. x deltaL = rec_cl_pair ( p , i , k ) deltaR = rec_cl_pair ( p , k + 1, j ) delta = min ( deltaL , deltaR ) // p [ i ], ... , p [ k ] is now sorted by y -coordinate, and // p [ k + 1], ... , p [ j ] is now sorted by y -coordinate. merge ( p , i , k , j ) // p [ i ], ... , p [ j ] is now sorted by y -coordinate. // store points in the vertical strip in v . t = 0 for k = i to j if ( p [ k ]. x > l - delta && p [ k ]. x < l + delta ) { t = t + 1 v [ t ] = p [ k ] } // look for closer pairs in the strip by comparing // each point in the strip to the next 7 points. for k = 1 to t - 1 for s = k + 1 to min ( t , k + 7) delta = min ( delta , dist ( v [ k ], v [ s ])) return delta }
- 11. Algorithm 5.4.1 Matrix Product This algorithm computes the product C of the n × n matrices A and B directly from the definition of the matrix product. Input Parameters: A , B Output Parameter: C matrix_product ( A , B , C ) { n = A . last for i = 1 to n for j = 1 to n { C [ i ][ j ] = 0 for k = 1 to n C [ i ][ j ] = C [ i ][ j ] + A [ i ][ k ] * B [ k ][ j ] } }