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keys in sequential order requires accessing about cix = 1 y entries for comparison 2 with the search argument The approaches described in this section reduce cix by replacing the sequential key structure of the entries in the current index block with a tree The use of rapid search in an index block regains computational time lost because of abbreviation We recall that the motivation for abbreviation is to increase the fanout ratio, which reduces the number of seek operations in the tree as well as the index space This section presents three alternate tree structures for indexes When entries are of xed size and not abbreviated, a conventional binary search or probing, as shown in Sec 3-2-3, also provide rapid access to entries within an index block A simple scheme groups the y index entries within one block into a number of sections to provide a second level The number of sections is chosen to be about equal to y An initial search pass jumps from section to section and compares one entry in each section A subsequent pass searches within the proper section through the y/ y = y entries found in a section The expected number of comparisons of sections and entries is 1 y each, so 2 that the total number of comparisons cix expected is jump search 4-4 cix = 2 ( 1 y) = y 2 which is the minimum obtainable with two passes and linear searching The num ber of comparisons y to nd an entry is within a factor of 2 of the optimum, a binary search, for typical values of y This additional pseudo-level does not require redundant entries of the key or pointers, only a means to locate the sections If the records are of xed length the location of the sections in the index block is computable Otherwise y section pointers can be used in each index block to de ne a two-level tree These section pointers have to be distinguishable so that they will not interfere with sequential processing of index entries An implementation is shown in Sec 4-3-3 and Fig 4-18.

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string inputString; inputString = Console.ReadLine( );

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Jump Index Search A Binary-Arranged Index Tree We presented the algorithm for binary search within a sequential le in Sec 3-2-1, Fig 3-5 Since entries in an index block are always sequential, a binary search can be used here too if the entries are of xed size For entries of arbitrary length a method which uses explicit successor pointers can permit a binary search to be used Figure 4-8 shows block D from the earlier example in Fig 4-7 The entries are arranged by level, so that the search begins at the middle value Two elds per entry direct the search algorithm to the next index entry to e ect a binary partitioning In this example, the maximum number of comparisons is four, and the expected number is 25/9 = 278 In a sequential search, (1 + 9)/2 = 5 comparisons would be expected In general, the tree search will require on the order of

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Whatever the user types at the console, until he presses Enter in other words, one line is assigned to inputString. It usually helps to give the user a prompt to let him know what he should be entering at a ReadLine( ), so it makes sense to start with a WriteLine( ):

TABLE: An arrangement of words, numbers, or signs, or combinations of them, as in parallel columns, to exhibit a set of facts or relations in a de nite, compact, and comprehensive form; a synopsis or scheme

binary search 4-5 cix = log2 y comparisons This is signi cant when there are many entries per index block The index shown has grown in size because of the pointers which relate the index entries to the tree structure Of these pointer elds, y + 1 will be empty In the example the comparison is based on the whole value of the key and some opportunities of high-order key abbreviation are lost

Console.WriteLine("Enter your input string.") inputString = Console.ReadLine( );

Computer systems are composed of many parts that are prone to failure, so that success in a given time period is achieved only if none of the parts fails For q parts having each a failure probability of pf the probability of successful operation is 11-1 psuccess = (1 pf )q We will look at the e ects of failures using some simple examples

Often, though, you ll want the user to enter his input on the same line, immediately after the prompt. In that case, you can use Write( ) instead of WriteLine( ). Write( ) outputs text to the console, the same as WriteLine( ), but it doesn t send a newline character. In other words, if you follow a Write( ) with a ReadLine( ), the cursor waits at the end of the output for the user s input:

If 10 parts, each with a probability of failure of 1% for a given operation, are used in a logically serial arrangement, the probability of achieving the correct result is

Console.Write("Enter your name here: "); string userName = Console.ReadLine( );

which amounts to a failure probability of nearly 10% For 100 such parts, the probability of system operation decreases to

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