computing the quartet distance between general trees

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28 CHAPTER 5. CALCULATING LEAF SET SIZESas indices. All entries associated with two edges pointing away from the root are filledby a recursive function. The function makes a combined depth-first traversal of the twotrees, compares the leaves and sums up the shared leaf set sizes on its way up through thetwo trees. Equation (5.1) shows that every time two subtrees are encountered there is adecision to make; for which subtree should we sum over the children? This can influencethe total number of additions the function has to make. Equation (5.2) includes thischoice as the min-expression, however, analysing the expression reveals that the choiceis not significant; the running time will stay within the asymptotic time bound whicheverway is chosen.After processing all pairs of subtrees not containing the root, the results are used tofill the remaining entries of the table, cf. Eq. 5.3.I have of course tested all parts of my code thoroughly for correctness and efficiency.However, as mentioned earlier, the running time of this procedure is critical to the overallrunning time of the cubic and sub-cubic algorithms and I have therefore decided ondocumenting the experiments verifying the time usage of the shared leaf set size calculation.The algorithm has been tested against each of the five types of test trees describedin Sec. 3.1.Expectations Before commenting on the results of my experiments I will discuss whichexpectations one might have to the actual performance of the algorithm. Clearly wewould expect it to perform within the theoretical time bounds of O(n 2 ), but what aboutthe actual time usage - will it e.g. perform equally good on every type of test tree or doesthe internal structure of a tree influence the running time?It is difficult to guess about the actual time spent by the algorithm. One thing to expect,however, is the C++ implementation to perform better than the Python implementation.With regards to the performance between different types of trees, one should takea look at the actual mechanisms of the algorithm. Where is the work really done? Whenlooking at Eq. (5.2) it is evident that the number of internal nodes is significant, meaningthat a high number of inner nodes will make the two sums very large. What about the degreeof inner nodes then? Even though there is a correspondence between the number ofinner nodes, |I |, and the degree of these, d v , as pointed out in Sec. 3.1, I find it hard to seethe exact influence on the algorithm here. From my point of view, it is easier to look atthe implementation details. Here I deal with directed edges pointing away from the rootand the number of these is clearly the same as the number of (undirected) edges in thetree altogether. From this point of view I would expect the number of edges to be criticalto the running time, of course meaning that fewer edges lead to a better performance.Therefore I expect binary trees, having 2n − 3 edges, to demand a high processing time,
5.3. IMPLEMENTING LEAF SET ALGORITHMS 29whereas the star tree, having n edges, would be faster to deal with. In between, sqrt-treesshould be faster than wc-trees, again due to the number of edges.This algorithm has a quadratic space consumption, due to the table being used forstorage, which should not cause any problems with regards to memory. As an example,think of two binary trees of 800 leaves. The table has an entry for each pair of subtrees,which is equal to each pair of directed edges. Since a binary tree has |E| = 2n − 3, theequation is roughly:2|E| × 2|E| × size_of_int ≈ 2 × 1600 × 2 × 1600 × 4 Bytes ≈ 41 MB.This will be no problem for the test environment described in Sec. 2.2, which will haveplenty of memory to spare.Result Figure 5.2 and 5.3 display the results of the two implementations, Python andC++ respectively, being applied to the five types of trees. The first thing to observe is thecorrectness of the time bound. Every plot is parallel to the line indicating a quadraticgrowth. Furthermore, the estimated exponents of the expressions describing the plotsare very close to 2. Next, the expectations about the order among the trees hold true.It seems to be correct that more edges lead to higher processing time. There is actuallyas large a difference as a factor of ten between the best and worst results. What is notas clear, because of the log-log-plot, is that the difference is more pronounced the largerthe trees become. This is the case, because the distance between plots remains the same,even though one step on an axis becomes increasingly significant when moving along theaxis. This is of course a consequence of the quadratic development.Last and less important, we can compare the two figures and see that the C++ implementationis clearly faster than the Python implementation.
Page 1: Master’s thesisCOMPUTING THE QUAR
Page 5: AcknowledgementsFirst of all I woul
Page 8 and 9: viiiCONTENTS5.3 Implementing leaf s
Page 10 and 11: 2 CHAPTER 1. INTRODUCTIONhas is cal
Page 12 and 13: 4 CHAPTER 1. INTRODUCTIONIt is evid
Page 14 and 15: 6 CHAPTER 1. INTRODUCTIONsub-cubic
Page 17 and 18: Chapter 2PrerequisitesFirst, this c
Page 19: 2.2. CHOICE OF LANGUAGE AND TEST EN
Page 22 and 23: 14 CHAPTER 3. EXPERIMENTAL APPROACH
Page 28 and 29: 20 CHAPTER 4. QUARTIC TIME ALGORITH
Page 30 and 31: 22 CHAPTER 4. QUARTIC TIME ALGORITH
Page 33 and 34: Chapter 5Calculating leaf set sizes
Page 35: 5.3. IMPLEMENTING LEAF SET ALGORITH
Page 39 and 40: Chapter 6Cubic time algorithmHere I
Page 41 and 42: 6.1. IMPLEMENTATION 33that we can l
Page 43 and 44: 6.1. IMPLEMENTATION 35carried out o
Page 45 and 46: Chapter 7Sub-cubic time algorithmIn
Page 47 and 48: 7.1. THE ALGORITHM 39CcabADFigure 7
Page 49 and 50: 7.1. THE ALGORITHM 41reflecting Eq.
Page 51 and 52: 7.1. THE ALGORITHM 43choice of indi
Page 53 and 54: 7.1. THE ALGORITHM 45More interesti
Page 55 and 56: 7.2. IMPLEMENTATION 477.2 Implement
Page 57 and 58: 7.2. IMPLEMENTATION 49tween the num
Page 59 and 60: 7.2. IMPLEMENTATION 51oretic approa
Page 61 and 62: 7.2. IMPLEMENTATION 53well. My impl
Page 63 and 64: 7.2. IMPLEMENTATION 55Performance o
Page 65: 7.2. IMPLEMENTATION 57ble sort, wil
Page 68 and 69: 60 CHAPTER 8. RESULTS AND DISCUSSIO
Page 71 and 72: Chapter 9ConclusionThe focus of thi
Page 73 and 74: Bibliography[1] Mukul S. Bansal, Ji
Page 75: BIBLIOGRAPHY 67[20] M.S. Waterman a
Page 78 and 79: 70 APPENDIX A. PREPROCESSING FOR TH
Page 81 and 82: Appendix CReal-life application of
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