110 lines
4.9 KiB
Plaintext
110 lines
4.9 KiB
Plaintext
Comparison Between BMZ And CHM Algorithms
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%!includeconf: CONFIG.t2t
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==Characteristics==
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Table 1 presents the main characteristics of the two algorithms.
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The number of edges in the graph [figs/img27.png] is [figs/img236.png],
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the number of keys in the input set [figs/img20.png].
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The number of vertices of [figs/img32.png] is equal
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to [figs/img12.png] and [figs/img237.png] for BMZ algorithm and the CHM algorithm, respectively.
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This measure is related to the amount of space to store the array [figs/img37.png].
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This improves the space required to store a function in BMZ algorithm to [figs/img238.png] of the space required by the CHM algorithm.
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The number of critical edges is [figs/img76.png] and 0, for BMZ algorithm and the CHM algorithm,
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respectively.
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BMZ algorithm generates random graphs that necessarily contains cycles and the
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CHM algorithm
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generates
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acyclic random graphs.
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Finally, the CHM algorithm generates [order preserving functions concepts.html]
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while BMZ algorithm does not preserve order.
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%!include(html): ''TABLE1.t2t''
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| **Table 1:** Main characteristics of the algorithms.
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----------------------------------------
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==Memory Consumption==
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- Memory consumption to generate the minimal perfect hash function (MPHF):
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|| Algorithm | //c// | Memory consumption to generate a MPHF |
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| BMZ | 0.93 | //24.80n + O(1)// |
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| BMZ | 1.15 | //26.42n + O(1)// |
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| CHM | 2.09 | //33.00n + O(1)// |
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| **Table 2:** Memory consumption to generate a MPHF using the algorithms BMZ and CHM.
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- Memory consumption to store the resulting minimal perfect hash function (MPHF):
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|| Algorithm | //c// | Memory consumption to store a MPHF |
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| BMZ | 0.93 | //3.72n// |
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| BMZ | 1.15 | //4.60n// |
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| CHM | 2.09 | //8.36n// |
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| **Table 3:** Memory consumption to store a MPHF generated by the algorithms BMZ and CHM.
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----------------------------------------
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==Run times==
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We now present some experimental results to compare the BMZ and CHM algorithms.
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The data consists of a collection of 100 million universe resource locations
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(URLs) collected from the Web.
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The average length of a URL in the collection is 63 bytes.
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All experiments were carried on
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a computer running the Linux operating system, version 2.6.7,
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with a 2.4 gigahertz processor and
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4 gigabytes of main memory.
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Table 4 presents time measurements.
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All times are in seconds.
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The table entries represent averages over 50 trials.
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The column labelled as [figs/img243.png] represents
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the number of iterations to generate the random graph [figs/img32.png] in the
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mapping step of the algorithms.
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The next columns represent the run times
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for the mapping plus ordering steps together and the searching
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step for each algorithm.
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The last column represents the percent gain of our algorithm
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over the CHM algorithm.
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%!include(html): ''TABLE4.t2t''
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| **Table 4:** Time measurements for BMZ and the CHM algorithm.
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The mapping step of the BMZ algorithm is faster because
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the expected number of iterations in the mapping step to generate [figs/img32.png] are
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2.13 and 2.92 for BMZ algorithm and the CHM algorithm, respectively
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(see [[2 bmz.html#papers]] for details).
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The graph [figs/img32.png] generated by BMZ algorithm
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has [figs/img12.png] vertices, against [figs/img237.png] for the CHM algorithm.
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These two facts make BMZ algorithm faster in the mapping step.
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The ordering step of BMZ algorithm is approximately equal to
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the time to check if [figs/img32.png] is acyclic for the CHM algorithm.
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The searching step of the CHM algorithm is faster, but the total
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time of BMZ algorithm is, on average, approximately 59 % faster
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than the CHM algorithm.
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It is important to notice the times for the searching step:
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for both algorithms they are not the dominant times,
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and the experimental results clearly show
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a linear behavior for the searching step.
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We now present run times for BMZ algorithm using a [heuristic bmz.html#heuristic] that
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reduces the space requirement
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to any given value between [figs/img12.png] words and [figs/img13.png] words.
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For example, for [figs/img244.png] and [figs/img6.png], the analytical expected number
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of iterations are [figs/img245.png] and [figs/img246.png], respectively
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(for [figs/img247.png], the number of iterations are 2.78 for [figs/img244.png] and 3.04
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for [figs/img6.png]).
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Table 5 presents the total times to construct a
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function for [figs/img247.png], with an increase from [figs/img248.png] seconds
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for [figs/img128.png] (see Table 4) to [figs/img249.png] seconds for [figs/img244.png] and
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to [figs/img250.png] seconds for [figs/img6.png].
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%!include(html): ''TABLE5.t2t''
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| **Table 5:** Time measurements for BMZ tuned algorithm with [figs/img5.png] and [figs/img6.png].
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%!include: ALGORITHMS.t2t
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%!include: FOOTER.t2t
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