Docstoc

A Study of Balanced Search Trees Brainstorming a New Balanced

Document Sample
A Study of Balanced Search Trees Brainstorming a New Balanced Powered By Docstoc
					A Study of Balanced Search Trees:
   Brainstorming a New Balanced Search Tree




            Anthony Kim, 2005

        Computer Systems Research
                    Abstract

This project investigates three different balanced
search trees for their advantages and disadvantages,
thus ultimately their efficiency. Run time and
memory space management are two main aspects
under the study. Statistical analysis is provided to
distinguish subtle differences if there is any. A new
balanced search tree is suggested and compared with
the four balanced search trees under study,.
Balanced search trees are implemented in C++
extensively using pointers and structs.
                  Introduction

● A simple binary search tree has some
disadvantages, specifically from its dependence on
the incoming data, that significantly affects its tree
structure hence its performance. (Ex. linear tree)
● An optimal search tree is one that tries to minimize

its height given some data. (Ex. Red-black tree has
2lg(n+1) height max)
● Some of balanced search trees are red-black tree,

AVL tree, weight-balanced tree, and B tree.
                  Background


● Simple vocabs: nodes, edges, children, parent, root
● Printing tree using recursion: pre-, in-, and post-

order traversal
● Basic binary search tree functions: insertion,

lookup, deletion (only first two apply in this project)
● Rotating functions: the key player in balancing

(Left rotation and right rotation.)
          Some Balanced Search
            (Red-black tree)
● Four properties
  ● The root of the tree is colored black.

  ● All paths from the root to the leaves agree on the

    number of black nodes.
  ● No path from the root to a leaf may contatin two

    consecutive nodes colored red.
  ● Every path from a node to a leaf (of the

    descendants) has the same number of black nodes
● Has height at most 2lg(n+1)
       Some Balanced Search Tree
         (Weight & height balanced search tree.)


● Weight balanced tree and height balanced tree are
very similar.
● Weight balanced tree (Height balanced tree) has

one property
  ● At each node, the difference between

    weight(height) of left subtree and weight(height)
    of right subtree is less than the threshold value.
●Supposedly yield height lg(n) at most
      A New Balanced Search Tree
                Median-weight-mix tree
● Assumption on statistical data
  ● Give lower bound and upper bound of total data

    input, random behavior is assumed, meaning data
    points will be evenly distributed throughout the
    interval
  ● Multiple “crests” is assumed to be present in the

    interval.
● Each node will have a key (data number), an

interval (with lower and upper bounds of its
assigned interval) and weights of left subtree and
right subtree.
        A New Balanced Search Tree
                  Median-weight-mix tree


●   Algorithm
    ● The weights of each subtree are calculated based

      on constants R and S
      ● R = the importance of focusing frequency heavy

        data points
      ● S = the importance of focusing frequency weak

        data points
    ● Left/right rotations to balanced R to S ratio
              Testing Methodology
●   14 Randomly generated test cases (test case size
    ranges 20 – 10,000)
●   4 Real test scores of math competition etc.
●   Things I am looking for
    –   Total Run Time
    –   Average Time Retrieval
    –   Height of Tree
    –   Average Retrieval Depth
Test Runs (Height-balanced Tree)
                                                                  Results
Total Run Time
                                                                         Average retrieval time
Test Case       Redblack       Height          Weight       MWM          Test Case       Redblack       Height              Weight          MWM
            1              0              0             0           0                1              0                  0               0                  0
            2              0              0             0           0                2              0                  0               0                  0
            3   N/A                       0             0           0                3   N/A                           0               0                  0
            4   N/A                     0.01            0           0                4   N/A                           0               0                  0
            5   N/A                     0.01        0.04            0                5   N/A                           0         0.000001                 0
            6   N/A                     0.01            0         0.01               6   N/A                           0               0                  0
            7   N/A                     0.01            0         0.02               7   N/A                     0.000002              0                  0
            8   N/A                     0.01        0.02          0.02               8   N/A                     0.000002              0            0.000004
            9   N/A                     0.02        0.02          0.02               9   N/A                           0               0                  0
        10      N/A                     0.03        0.02          0.05            10     N/A                     0.000001        0.000002           0.000001
        11      N/A                     0.03        0.04          0.05            11     N/A                     0.000002        0.000001           0.000001
        12      N/A                     0.03        0.04          0.05            12     N/A                     0.000002        0.000002           0.000001
        13      N/A                     0.04        0.04          0.05            13     N/A                     0.000002        0.000002           0.000001
        14      N/A                     0.03        0.04          0.05            14     N/A                     0.000001        0.000002                 0
       101                 0              0             0           0            101                0                  0               0                  0
       102                 0              0             0           0            102                0                  0               0                  0
       103      N/A                     0.08        0.41    N/A                  103     N/A                 0.0000007         0.00000086   N/A
       104             0.02             0.02        0.05          0.02           104        0.000181        0.00000089                 0          0.00000093
                                                                      Results
Depth
                                                                           Average Retrieval Depth

Test Case         Redblack          Height        Weight        MWM        Test Case        Redblack         Height             Weight             MWM
                                                                                        1              3.5               2.35               2.35            2.35
              1                 7             4             4          4
                                                                                        2            5.68                 3.9               5.08            4.42
              2                12             6             9          7
                                                                                        3   N/A                          4.85               5.44            5.31
              3   N/A                         8            10         10
                                                                                        4   N/A                          7.01              8.684           7.422
              4   N/A                        11            16         13
                                                                                        5   N/A                         8.217             11.823           8.821
              5   N/A                        12            24         14
                                                                                        6   N/A                        9.2445             13.721          9.9635
              6   N/A                        13            29         17
                                                                                        7   N/A                       10.3778             14.674         12.0244
              7   N/A                        14            31         20                8   N/A                       10.4804              14.52         11.6468
              8   N/A                        14            31         19                9   N/A                       10.4662            15.1646         11.6474
              9   N/A                        15            38         18               10   N/A                       11.5749            18.5899         13.2171
             10   N/A                        16            38         23               11   N/A                       11.5864            15.6778         13.0158

             11   N/A                        16            37         22               12   N/A                       11.6456            16.4298         13.1199

             12   N/A                        16            37         22               13   N/A                       11.6748            16.5699         13.3742

             13   N/A                        16            36         23               14   N/A                       11.5393            17.7741         13.1296
                                                                                   101            49.0291             4.87379            8.50971         5.18932
             14   N/A                        16            33         24
                                                                                   102            18.5625                 4.4               4.55          4.7375
            101               101             7            18          9
                                                                                   103      N/A                   0.000765942       0.00065652     N/A
            102                38             6             8          7
                                                                                   104            2368.98             1.68751            2.70135         2.62517
            103   N/A                        16            42   N/A

            104              5596            13            24         12
                       Result
●   Total run time and average retrieval time data did
    not make any sense.
●   Hard to time processes on fast computers
●   Red-black tree segmentation faulted for large test
    cases >500, so it provided no experimental data
                         Result (Height)
                                 Height

         40
                                 y = 5.3315Ln(x) - 12.735
         35
         30
         25                         y = 2.9492Ln(x) - 5.1306
Height




         20
         15
         10                          y = 1.8939Ln(x) - 1.517
          5
          0
              0   2000    4000        6000      8000        10000   12000
                                    Data size


                           Height     Weight    MWM
Result (Average retrieval depth)
                        Average Retrieval Depth

        20
                 y = 2.3718Ln(x) - 4.9972
        15                             y = 1.7446Ln(x) - 3.0532
Depth




        10
                                     y = 1.4998Ln(x) - 2.2646

        5

        0
             0     2000       4000       6000      8000         10000   12000
                                      Data size


                          Redblack    Height    Weight    MWM
                      Analysis
●   All balanced search trees show logarithmic
    characteristics for height and average retrieval
    depth as expected. (except red-black tree)
●   Height-balanced tree seems to perform the best
    among three working balanced search trees.
●   Median-weight-mix tree’s logarithmic line lies
    between height-balanced tree’s line and weight-
    balanced tree’s line.
                    Conclusion
●   The project experimentally showed that balanced
    binary search trees show logarithmic
    characteristics.
●   Median-weight-mix tree’s performance is an
    intermediate between height-balanced tree’s and
    weight-balanced tree’s.
●   More studies should be done on other balanced
    search trees or variants of search trees studied in
    this project

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:0
posted:6/19/2013
language:English
pages:17