A Practical Introduction to Data Structures and Algorithm Analysis

342 Chap. 9 Searching09050090501100111001223344556679877798778203782037991059(a)(b)Figure 9.6 Example of problems with linear probing. (a) Four values are insertedin the order 1001, 9050, 9877, and 2037 using hash function h(K) = K mod 10.(b) The value 1059 is added to the hash table.in slot 2 with probability 6/10. This is illustrated by Figure 9.6(b). This tendencyof linear probing to cluster items together is known as primary clustering. Smallclusters tend to merge into big clusters, making the problem worse. The objectionto primary clustering is that it leads to long probe sequences.Improved Collision Resolution MethodsHow can we avoid primary clustering? One possible improvement might be to uselinear probing, but to skip slots by a constant c other than 1. This would make theprobe functionp(K, i) = ci,and so the ith slot in the probe sequence will be (h(K) + ic) mod M. In this way,records with adjacent home positions will not follow the same probe sequence. Forexample, if we were to skip by twos, then our offsets from the home slot wouldbe 2, then 4, then 6, and so on.One quality of a good probe sequence is that it will cycle through all slots inthe hash table before returning to the home position. Clearly linear probing (which“skips” slots by one each time) does this. Unfortunately, not all values for c willmake this happen. For example, if c = 2 and the table contains an even numberof slots, then any key whose home position is in an even slot will have a probe