The document describes binary search, an algorithm for finding the index of a target value in a sorted array. It maintains the invariant that the value is between the lower and upper bounds. The example performs a binary search for the value 33 in a sample array.

1. 8
2
1 3 4 6
5 7 10
9 11 12 14
13
0
64
14
13 25 33 51
43 53 84
72 93 95 97
96
6
Binary Search
lo
Binary search. Given value and sorted array a[], find index i
such that a[i] = value, or report that no such index exists.
Invariant. Algorithm maintains a[lo]  value  a[hi].
Ex. Binary search for 33.
hi

2. Binary Search
Binary search. Given value and sorted array a[], find index i
such that a[i] = value, or report that no such index exists.
Invariant. Algorithm maintains a[lo]  value  a[hi].
Ex. Binary search for 33.
8
2
1 3 4 6
5 7 10
9 11 12 14
13
0
64
14
13 25 33 51
43 53 84
72 93 95 97
96
6
lo hi
mid

3. Binary Search
Binary search. Given value and sorted array a[], find index i
such that a[i] = value, or report that no such index exists.
Invariant. Algorithm maintains a[lo]  value  a[hi].
Ex. Binary search for 33.
8
2
1 3 4 6
5 7 10
9 11 12 14
13
0
64
14
13 25 33 51
43 53 84
72 93 95 97
96
6
lo hi

4. Binary Search
Binary search. Given value and sorted array a[], find index i
such that a[i] = value, or report that no such index exists.
Invariant. Algorithm maintains a[lo]  value  a[hi].
Ex. Binary search for 33.
8
2
1 3 4 6
5 7 10
9 11 12 14
13
0
64
14
13 25 33 51
43 53 84
72 93 95 97
96
6
lo mid hi

5. Binary Search
Binary search. Given value and sorted array a[], find index i
such that a[i] = value, or report that no such index exists.
Invariant. Algorithm maintains a[lo]  value  a[hi].
Ex. Binary search for 33.
8
2
1 3 4 6
5 7 10
9 11 12 14
13
0
64
14
13 25 33 51
43 53 84
72 93 95 97
96
6
lo hi

6. Binary Search
Binary search. Given value and sorted array a[], find index i
such that a[i] = value, or report that no such index exists.
Invariant. Algorithm maintains a[lo]  value  a[hi].
Ex. Binary search for 33.
8
2
1 3 4 6
5 7 10
9 11 12 14
13
0
64
14
13 25 33 51
43 53 84
72 93 95 97
96
6
lo hi
mid

7. Binary Search
Binary search. Given value and sorted array a[], find index i
such that a[i] = value, or report that no such index exists.
Invariant. Algorithm maintains a[lo]  value  a[hi].
Ex. Binary search for 33.
8
2
1 3 4 6
5 7 10
9 11 12 14
13
0
64
14
13 25 33 51
43 53 84
72 93 95 97
96
6
lo
hi

8. Binary Search
Binary search. Given value and sorted array a[], find index i
such that a[i] = value, or report that no such index exists.
Invariant. Algorithm maintains a[lo]  value  a[hi].
Ex. Binary search for 33.
8
2
1 3 4 6
5 7 10
9 11 12 14
13
0
64
14
13 25 33 51
43 53 84
72 93 95 97
96
6
lo
hi
mid

9. Binary Search
Binary search. Given value and sorted array a[], find index i
such that a[i] = value, or report that no such index exists.
Invariant. Algorithm maintains a[lo]  value  a[hi].
Ex. Binary search for 33.
8
2
1 3 4 6
5 7 10
9 11 12 14
13
0
64
14
13 25 33 51
43 53 84
72 93 95 97
96
6
lo
hi
mid