exercises show the result of the following sequence of instructionsunion( , )union( , )union( , )union( , )union( , )union( , )union( , )union( , )union ( , )union( , )union( , )union( , )union( , )union( , )union ( , )union( when the unions are performed arbitrarily performed by height performed by size for each of the trees in the previous exerciseperform find with path compression on the deepest node write program to determine the effects of path compression and the various unioning strategies your program should process long sequence of equivalence operations using all six of the possible strategies show that if unions are performed by heightthen the depth of any tree is (log suppose (nis nicely defined function that reduces to smaller integer what ( ( ) with appropriate initial is the solution to the recurrence (nf(nconditions show that if then the running time of union/find operations is (mb show that if log nthen the running time of union/find operations is (mc suppose ( log log nwhat is the running time of union/find operationsd suppose ( lognwhat is the running time of union/find operationstarjan' original bound for the union/find algorithm defined (mnmin{ > |( (im/ log )}where ( >= ( ( (ija( (ij ) >= ij > herea(mnis one version of the ackermann function are the two definitions of asymptotically equivalentprove that for the mazes generated by the algorithm in section the path from the starting to ending points is unique design an algorithm that generates maze that contains no path from start to finish but has the property that the removal of prespecified wall creates unique path suppose we want to add an extra operationdeunionwhich undoes the last union operation that has not been already undone show that if we do union-by-height and finds without path compressionthen deunion is easyand sequence of unionfindand deunion operations takes ( log ntime why does path compression make deunion hard