Analysis of Searching Algorithms

Transcription

1 CS 4/56101 Design and Analysis of Algorithms Kent State University Dept. of Math & Computer Science LECT-4 2 Analysis of Searching Algorithms 3 1

2 Search LECT-04, S-4 Sequential Search int SequentialSearch(List list, KeyType target) int location; for (location = 0; location < list.count; location++) if (EQ(list.entry[location].key, target)) return location; return -1; LECT-04, S-5 2

3 Binary Search 6 Binary Search The method date back at least to 1946, but the first version free of errors and unnecessary restrictions seems to appeared only in 1962! One study showed that about 90% of professional programmers fail to code binary search correctly, even after working on it one full hour. LECT-04, S-7 3

4 Idea Start with an ordered list In searching an ordered list: first compare the target to the key in the center of the list. If it is smaller, restrict the search to the left half; otherwise restrict the search to the right half, and repeat. In this way, at each step we reduce the length of the list to be searched by half. < target >= target bottom top LECT-04, S-8 Binay1Search /* RecBinary1Search: front end for RecBinary1. */ int RecBinary1Search(List list, KeyType target) return RecBinary1(list, target, 0, list.count-1); LECT-04, S-9 4

5 Binary1Search int RecBinary1(List list, KeyType target, int bottom, int top) int middle = -1; if (bottom < top) /* The list has size greater than 1. */ middle = (top + bottom) / 2; if (GT(target, list.entry[middle].key)) /* Reduce to the top half of the list.*/ middle = RecBinary1(list, target, middle+1, top); else /* Reduce to the bottom half of the list.*/ middle = RecBinary1(list, target, bottom, middle); else if (bottom == top) /* The list has exactly 1 entry. */ if (EQ(target, list.entry[top].key)) middle = top; return middle; bottom bottom middle <target top middle >= target top >= target >=target top >= target Middle=10 target=12 target=7 Target=10 LECT-04, S-10 Binary Search with Equality Check Binary1 may make unnecessary iteration, because it may fail to recognize that middle is the target! LECT-04, S-11 5

6 Binary2 Search /* RecBinary2Search: front end for RecBinary2. */ int RecBinary2Search(List list, KeyType target) return RecBinary2(list, target, 0, list.count-1); LECT-04, S-12 Recursive Binary2 int RecBinary2(List list, KeyType target, int bottom, int top) int middle = -1; if (bottom <= top) middle = (top + bottom) / 2; if (LT(target, list.entry[middle].key)) /* Reduce to the bottom half.*/ middle = RecBinary2(list,target,bottom,middle-1); else if (GT(target, list.entry[middle].key) /* Reduce to the top half.*/ middle = RecBinary2(list, target, middle + 1, top); return middle; LECT-04, S-13 6

7 Analysis: Which Version is Better (Vote) 14 Comparison Tree The comparison tree of an algorithm is obtained by tracing the action of the algorithm, representing each comparison of keys by a vertex of the tree (which we draw as a circle). LECT-04, S-15 7

8 Comparison Tree for Binary1 Search (with 10 keys) Each node represents 1 comparison. Height represents worst cast running time. LECT-04, S-16 Comparison Tree for Binary1 Search (with 10 keys) External Path Length = (4x5)+(6x4)+(4x5)+(6x4)=88 Half of the paths are successful half of them are unsuccessful. Average path length = 44/10=4.4 for both cases. LECT-04, S-17 8

9 Comparison Tree for Binary2Search with n=10 keys (compact drawing) Each node, except for the last successful one, represents 2comparisons. Height represents worst cast running time. LECT-04, S-18 Comparison Tree for Binary2Search with n=10 keys (compact drawing) Average successful search: 1+(2x3)+(4x5)+(3x7)=48; 48/10=4.8 Average unsuccessful search: 6x5+8x6=78; 78/ LECT-04, S-19 9