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data are shown in Fig. 12.6. The rst 1000 frames are shown on the left. The test values identify the scene changes with no false positives. The choice of 0:5 as the critical value is not signi cant. Frames 9001 through 10,000 are shown on the right. This is a more active (in terms of bit-rate uctuations) subsequence, and changing the critical value will affect the number of frames identi ed as scene changes. There are 317 scene changes when 0:5 is the critical value (the mean scene length is 6.5 seconds); there are 374 when 0:4 is used, and 283 when 0:6 is used. The density functions of the scene lengths produced by these critical values are shown in Fig. 12.7. We observe that the critical value does not have a large effect on the density function. 12.3.2.1 Scene Lengths Plots of the autocorrelation function showed that scene lengths are uncorrelated, so the main modeling issue is to characterize the distribution of the number of frames in a scene. The shape of the density in Fig. 12.7 was observed on all sequences except for news. (This is a news broadcast. There are 134 scene changes, and 75 of them are 3 frames long. Moreover, most of these occur consecutively. We do not have an explanation for this. The 3-frame scenes were deleted and the remainder are called news.d.) This unimodal and long-tailed shape is characteristic of distributions used in reliability and insurance-loss models. We will use the following three distributions as candidates for describing scene lengths (and the number of cells per frame in the sequel).



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The Gamma Distribution. The density function is given by Eq. (12.3). The Weibull Distribution. It has the complementary distribution function (the probability that the random variable is larger than x) SW x 1 e lx ;





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77 EXERCISES 71 Draw the network of the following parallel systems as a topology graph (De nition 73): (a) A cluster of PCs composed of three subnetworks In each subnetwork 8 processors are connected to one central switch The networks are linked through a direct connection from each switch to every other switch (b) A system of 9 processing nodes, which are connected through a 3 3 cyclic grid (Section 221) Each processing node consists of one interface to the grid and two processors, all connected via a single bus Describe how to model a butter y network of 8 processors (depicted in Figure 211) as a topology graph Consider: (a) The edge types to be employed, that is, directed, undirected, or hyperedge (b) Can the communication routes in such a topology graph be determined with a shortest-path algorithm or is a speci c routing algorithm necessary Give examples for routes between processors (c) Draw the topology graph

x; l; b > 0:

The Generalized Pareto Distribution. It has density function fP x G b k lb xk 1 G b G k l x k b ; x; b; l; k > 0:

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The Classical Pareto Distribution is the special case of k 1. We will dispense with the adjective ``general.'' In all of these distributions, l is a scale parameter and b is a shape parameter; the Pareto distribution has the second shape parameter k. The Pareto distribution contains the other two as special cases, but it is useful to maintain the distinction. Figure 2.1 of Cox and Oakes [6] gives a nice way to compare these distributions. For a random variable X , let m E X , s2 Var X , and m3 E X m 3 . Let g1 be the coef cient of variation; that is, g1 s=m. Let g3 m3 =s3 ; it is the coef cient of skewness. For the distributions described above (and some others not mentioned here), g1 and g3 do not depend on the scale parameter l. Plotting g3 versus g1 gives a

73 Visit the online Overview of Recent Supercomputers by van der Steen and Dongarra [193] on the site of the TOP500 Supercomputer Sites [186] and choose one modern system Model this system with a topology graph (De nition 73) 74 Figure 79 displays the topology graph for 8 dual processors connected via a switch:

curve for the gamma and Weibull distributions, and a curve for each value of k for the Pareto distribution. Figure 12.8 shows these curves, and points for the sample values for the scene-length moments from our video sequences. (We excluded bingo from this and all subsequent gures because it contained only 12 scenes. We included data from a software intraframe coding [10] of the lm ``Star Wars.'') Except for sequences news and news.d, the moments of all the sequences plotted fall near the four curves traced out by the distributions; the distances from the curves are within the deviations we observed when samples from a true gamma distribution were compared with the gamma curves. We conclude that usually, but not always, scene lengths will follow a unimodal distribution that ts one of the common failuretime distributions. 12.3.3 Scene-Change Frames

Java is very careful in how it handles Strings in order to minimize storage requirements and increase performance. In particular, Strings are immutable (that is, once a String is created it cannot be modi ed) and the VM attempts to ensure that there is only one copy of any string in the String literal pool. This section outlines a number of issues that arise from this approach.

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