We are going to derive an upper bound for the average number of exchanges for quicksort. A similar analysis would give a lower bound, giving the high order term exactly. (a) Assume that the partition (or pivot) element ends up in position q. How many exchanges does partition do, NOT counting the final exchange where the pivot element is placed in its proper sorted position? Briefly justify. Note that an element can exchange with itself. (b) Write a recurrence for the expected number of exchanges (for quicksort), NOT counting the final exchange where pivot element is placed in its proper sorted position. (c) Simplify the recurrence as much as reasonably possible (as we did in class for comparisons). (d) Guess that the solution is at most an ln n for some constant a. Use constructive induction to verify the guess and derive the constant a. (e) Give an upper bound on how many exchanges involve the pivot element thoughout all of the partitions in the entire quicksort algorithm. Briefly justify.(f) Add this value to your answer in Part (d) to get an upper bound on the total number of exchanges. (g) Rewrite your solution using log base 2 rather than the natural log, evaluating the constant to three decimal places.

Program using c++ only on visual studio pig is a simple two player dice game, played with one die. the first player to reach or surpass 50 is the winner. each player takes a turn rolling the dice. they add to the pot with each roll, having to decide to roll again and increase the pot, or cash out. the risk being they could lose the amount they’ve accumulated into the pot. the rules for each player’s die roll. 1. roll the dice. a. if user rolled a 1, i. the pot gets set to zero ii. the other player goes to step 1. b. a roll of 2-6 is added to the pot. 2. user can choose to hold or roll again. a. choice roll. return to step 1. b. choice hold. i. increment player score by the pot amount. ii. pot gets set to 0. iii. second player gets to roll and goes to step 1. program requirements: ● before each opponent begins ○ output the score for the person and the computer. ○ output the opponents whose turn is beginning and ask the user to hit enter to continue. ● with each dice roll. ○ output the die value, and amount of the round pot. ○ if it’s the users roll ask if they want to roll again ( r ) or hold ( h ). your program should allow r, r, h or h as valid input. if input is anything else, ask the user again until valid input is obtained. ○ the ai will continue playing until the round pot is 20 or more. ● once a player’s score is greater or equal to 50 then they have won, it will no longer ask if they want to keep rolling the die or not. ● once there is a winner ○ score totals are output along with who the winner was. user or computer ○ player is asked if they want to play again y or n. valid input should be y, y, or n, n. ● when a new game starts the starting roll goes to the player that did not roll last. if the user rolled last in the previous game, then the computer rolls first and vice versa. when the program first begins, the player will make the first roll of the first game. development notes : ● you will need a way to roll dice in your program. the rand() function works well, but returns an integer. if we want numbers 0 – 9 we can get the value modulus 10. ● call srand() with a value to seed it. it’s common to seed it with the current computer clock, include ctime, and then call srand(time(

The editing of digital photos us about the same level of difficulty as editing an analog photo

Which best explains why a digital leader would join a society specializing in technology