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Chapter: Introduction to the Design and Analysis of Algorithms : Brute Force and Exhaustive Search

Sequential Search and Brute-Force String Matching

The first deals with the canonical problem of searching for an item of a given value in a given list. The second is different in that it deals with the string-matching problem.

Sequential Search and Brute-Force String Matching


We saw in the previous section two applications of the brute-force approach to the sorting porblem. Here we discuss two applications of this strategy to the problem of searching. The first deals with the canonical problem of searching for an item of a given value in a given list. The second is different in that it deals with the string-matching problem.


Sequential Search


We have already encountered a brute-force algorithm for the general searching problem: it is called sequential search (see Section 2.1). To repeat, the algorithm simply compares successive elements of a given list with a given search key until either a match is encountered (successful search) or the list is exhausted without finding a match (unsuccessful search). A simple extra trick is often employed in implementing sequential search: if we append the search key to the end of the list, the search for the key will have to be successful, and therefore we can eliminate the end of list check altogether. Here is pseudocode of this enhanced version.




ALGORITHM                 SequentialSearch2(A[0..n], K)


//Implements sequential search with a search key as a sentinel //Input: An array A of n elements and a search key K


//Output: The index of the first element in A[0..n 1] whose value is


            equal to K or 1 if no such element is found


A[n] K i 0


while A[i]  = K do i i + 1


if i < n return i else return 1


Another straightforward improvement can be incorporated in sequential search if a given list is known to be sorted: searching in such a list can be stopped as soon as an element greater than or equal to the search key is encountered.


Sequential search provides an excellent illustration of the brute-force ap-proach, with its characteristic strength (simplicity) and weakness (inferior effi-ciency). The efficiency results obtained in Section 2.1 for the standard version of sequential search change for the enhanced version only very slightly, so that the algorithm remains linear in both the worst and average cases. We discuss later in the book several searching algorithms with a better time efficiency.


 Sequential Search and Brute-Force String Matching


Brute-Force String Matching


Recall the string-matching problem introduced in Section 1.3: given a string of n characters called the text and a string of m characters (m n) called the pattern, find a substring of the text that matches the pattern. To put it more precisely, we want to find i—the index of the leftmost character of the first matching substring in the text

If matches other than the first one need to be found, a string-matching algorithm can simply continue working until the entire text is exhausted.


A brute-force algorithm for the string-matching problem is quite obvious: align the pattern against the first m characters of the text and start matching the corresponding pairs of characters from left to right until either all the m pairs of the characters match (then the algorithm can stop) or a mismatching pair is encountered. In the latter case, shift the pattern one position to the right and resume the character comparisons, starting again with the first character of the pattern and its counterpart in the text. Note that the last position in the text that can still be a beginning of a matching substring is n m (provided the text positions are indexed from 0 to n 1). Beyond that position, there are not enough characters to match the entire pattern; hence, the algorithm need not make any comparisons there.



ALGORITHM     BruteForceStringMatch(T [0..n 1], P [0..m 1])


//Implements brute-force string matching


//Input: An array T [0..n 1] of n characters representing a text and


            an array P [0..m 1] of m characters representing a pattern //Output: The index of the first character in the text that starts a


            matching substring or 1 if the search is unsuccessful


for i 0 to n m do j 0


while j < m and P [j ] = T [i + j ] do j j + 1


if j = m return i return 1


An operation of the algorithm is illustrated in Figure 3.3. Note that for this example, the algorithm shifts the pattern almost always after a single character comparison. The worst case is much worse: the algorithm may have to make all m comparisons before shifting the pattern, and this can happen for each of the n m + 1 tries. (Problem 6 in this section’s exercises asks you to give a specific example of such a situation.) Thus, in the worst case, the algorithm makes

FIGURE 3.3 Example of brute-force string matching. The pattern’s characters that are compared with their text counterparts are in bold type.



m(n m + 1) character comparisons, which puts it in the O(nm) class. For a typical word search in a natural language text, however, we should expect that most shifts would happen after very few comparisons (check the example again). Therefore, the average-case efficiency should be considerably better than the worst-case efficiency. Indeed it is: for searching in random texts, it has been shown to be linear, i.e.,  (n). There are several more sophisticated and more efficient algorithms for string searching. The most widely known of them—by R. Boyer and J. Moore—is outlined in Section 7.2 along with its simplification suggested by R. Horspool.


Exercises 3.2


Find the number of comparisons made by the sentinel version of sequential search in the worst case.

 in the average case if the probability of a successful search is p (0 p 1).


As shown in Section 2.1, the average number of key comparisons made by sequential search (without a sentinel, under standard assumptions about its inputs) is given by the formula

where p is the probability of a successful search. Determine, for a fixed n, the values of p (0 p 1) for which this formula yields the maximum value of Cavg(n) and the minimum value of Cavg(n).


Gadget testing A firm wants to determine the highest floor of its n-story headquarters from which a gadget can fall without breaking. The firm has two identical gadgets to experiment with. If one of them gets broken, it cannot be repaired, and the experiment will have to be completed with the remaining gadget. Design an algorithm in the best efficiency class you can to solve this problem.

            Determine the number of character comparisons made by the brute-force algorithm in searching for the pattern GANDHI in the text




Assume that the length of the text—it is 47 characters long—is known before the search starts.


            How many comparisons (both successful and unsuccessful) will be made by the brute-force algorithm in searching for each of the following patterns in the binary text of one thousand zeros?


a. 00001          b. 10000 c. 01010


            Give an example of a text of length n and a pattern of length m that constitutes a worst-case input for the brute-force string-matching algorithm. Exactly how many character comparisons will be made for such input?


            In solving the string-matching problem, would there be any advantage in comparing pattern and text characters right-to-left instead of left-to-right?


            Consider the problem of counting, in a given text, the number of substrings that start with an A and end with a B. For example, there are four such substrings in CABAAXBYA.


            Design a brute-force algorithm for this problem and determine its effi-ciency class.


            Design a more efficient algorithm for this problem. [Gin04]


            Write a visualization program for the brute-force string-matching algorithm.


            Word Find A popular diversion in the United States, “word find” (or “word search”) puzzles ask the player to find each of a given set of words in a square table filled with single letters. A word can read horizontally (left or right), vertically (up or down), or along a 45 degree diagonal (in any of the four directions) formed by consecutively adjacent cells of the table; it may wrap around the table’s boundaries, but it must read in the same direction with no zigzagging. The same cell of the table may be used in different words, but, in a given word, the same cell may be used no more than once. Write a computer program for solving this puzzle.


Battleship game Write a program based on a version of brute-force pattern matching for playing the game Battleship on the computer. The rules of the game are as follows. There are two opponents in the game (in this case, a human player and the computer). The game is played on two identical boards (10 × 10 tables of squares) on which each opponent places his or her ships, not seen by the opponent. Each player has five ships, each of which occupies a certain number of squares on the board: a destroyer (two squares), a submarine (three squares), a cruiser (three squares), a battleship (four squares), and an aircraft carrier (five squares). Each ship is placed either horizontally or vertically, with no two ships touching each other. The game is played by the opponents taking turns “shooting” at each other’s ships. The result of every shot is displayed as either a hit or a miss. In case of a hit, the player gets to go again and keeps playing until missing. The goal is to sink all the opponent’s ships before the opponent succeeds in doing it first. To sink a ship, all squares occupied by the ship must be hit.

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Introduction to the Design and Analysis of Algorithms : Brute Force and Exhaustive Search : Sequential Search and Brute-Force String Matching |

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