last lab

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lab10/lab10

+Q1: 
+
+in the table, wherever stated f(n) = xxxx, it means the data in that line is T(n) /f(n)
+
+T(n): seems to be increasing til infinity --> bound below loosely by a constant
+f(n) = n: T(n)/n  seems to be increasing to infinity --> it is bound below loosely by a line
+f(n) = n^2: it is moving towards 0, although it did slows down but the trend is unlikely to reach a constant --> can say that it is bound above lightly by n2 although tighter than n^3 
+f(n) = n^3: seems to be decreasing too --> bound loosely above by a n^3 function
+
+f(n) = nlog(n) seems to be the one with trend approaching a constant fastest ==> can say that T(n) grows similarly to a nlog(n) function
+
+    
+     n           T(n)    f(n)=n*10^5   f(n)=nlog(n)  f(n)=n^2*10^9  f(n)=log(n)*10^5     f(n)=n^3*10^6
+  1000         0.1900        19.0000        27.5053       190.0000         2750.5317           19.0000
+ 10000         2.2400        22.4000        24.3205        22.4000        24320.4910            0.2240
+ 20000         4.1100        20.5500        20.7503        10.2750        41500.5318            0.0514
+ 30000         5.8000        19.3333        18.7539         6.4444        56261.7774            0.0215
+ 40000         7.9000        19.7500        18.6380         4.9375        74551.9705            0.0123
+ 50000        11.2200        22.4400        20.7398         4.4880       103698.9826            0.0090
+ 60000        14.2600        23.7667        21.6019         3.9611       129611.6215            0.0066
+ 70000        15.0400        21.4857        19.2589         3.0694       134812.3186            0.0044
+ 80000        20.8100        26.0125        23.0407         3.2516       184325.9698            0.0041
+
+
+ 
+Q2:
+f(n) = n gives the numbers of swaps goes to nfinity as n goes to infinity ---> it is a lower bound
+f(n) = nlog(n) was pretty close to being the theta to T(n) but it still has the decrease trend --> upper bound but much tighter than n^2
+f(n) = n^2 decreases fast towards 0 --> loose upper bound
+f(n) = log(n) increases --> lower bound
+
+     n			  T(n)		 f(n) = n		f(n) = nlog(n)	    f(n) = n^2	   f(n) = log(n)	  f(n) = n^2log(n)
+  1000		  3425457		 3425.457		      1141.819			3.425457		      1141819		    0.168930947   
+ 10000		 45721062		4572.1062	       1143.02655		 0.45721062	      11430265.5		    0.114387598
+ 20000		 98107982		4905.3991	      1140.517296		0.245269955	     22810345.92		    0.057025093
+ 30000		153522280	 5117.409333	      1143.013343		0.170580311	      34290400.3		    0.038183995
+ 40000	   210173451	 5254.336275	      1141.735719		0.131358407	     45669428.78		    0.028994489
+ 50000		265180908	  5303.61816 	      1128.676743		0.106072363	     56433837.15		    0.022509831
+ 60000		319847373	  5330.78955	       1115.65944		0.088846493	     66939566.42		    0.018502093
+
+ Through these 2 tables, the data still show that the quicksort program still grows most similar with theta(nlog(n)) despite the differencce in the method of comparison: time measurement and counting number of swap function calls.