**Data Structures and Algorithm Analysis**

1. How much memory will the following structures take up? Hint: int takes 4 bytes, float takes 4 bytes, char takes 1 byte. – 9 points

a. struct Structure1 {int a,b,c,d,e[10]; float f;};

b. struct Structure2 {int a[12]; float b[5];};

c. struct Structure3 {char a[10][12][4], b[5][5], c[10];};

2. Show the steps when sorting (smallest to largest) the following arrays of numbers using **selection sort** i.e. every time a swap occurs, show what the current state of the array looks like, including the final sorted array. – 12 points

a. [10, 2, 5, 8, 9, 1, 4, 7]

b. [7, 1, 3, 2, 5, 4, 8, 12, 9]

c. [8, 7, 6, 5, 4, 3, 2, 1]

d. [5, 3, 8, 1, 9, 4, 2, 6]

3. Big-O: What is meant by f(n) = O(g(n))? Give the definition and then explain what it means in your own terms. – 5 points

4. Big-Omega: What is meant by f(n) = Ω(g(n))? Give the definition and then explain what it means in your own terms. – 5 points

5. Show that , make sure you use the definition and justify the inequalities and constants used. – 4 points

6. Show that , make sure you use the definition and justify the inequalities and constants used. – 4 points

7. Show that , make sure you use the definition and justify the inequalities and constants used. – 4 points

8. Show that , make sure you use the definition and justify the inequalities and constants used. – 4 points

9. Show that , make sure you use the definition and justify the inequalities and constants used. – 4 points

10. What principle governs how you add/remove elements in a stack? Spell it out and briefly explain. – 4 points

11. Briefly describe an application of a stack. – 4 points

12. What principle governs how you add/remove elements in a queue? Spell it out and briefly explain. – 4 points

13. Briefly describe an application of a queue. – 4 points

Consider the following graph (pseudocode for BFS and DFS given on page 9):

14. Write the order in which the nodes would be visited in when doing a **breadth first search (BFS)** traversal starting at **node 4**. Also, write the distances from 4 to every other node. – 6 points

15. Write the order in which the nodes would be visited in when doing a **breadth first search (BFS)** traversal starting at **node 5**. Also, write the distances from 5 to every other node. – 6 points

Same graph (for your convenience):

16. Write the order in which the nodes would be visited in when doing a **depth first search (DFS)** traversal starting at **node 4 (order discovered or order off the stack)**. – 6 points

17. Write the order in which the nodes would be visited in when doing a **depth first search (DFS)** traversal starting at **node 5 (order discovered or order off the stack)**. – 6 points

18. Give the definition of a graph. – 5 points

19. Give the definition of a tree (from graph theory). – 4 points

**BFS Pseudocode (for graph with n vertices):**

Input: grapharray[n][n], source

queue<int> Q

int distance[n] (array to keep track of nodes distances (from source), all values set to -1 except source which is set to 0 i.e. -1 = not visited)

Q.push(source)

while(Q is not empty)

v = Q.front

Q.pop()

for each neighbor w of v

if distance[w] = -1

print w

distance[w] = distance[v]+1

Q.push(w)

end if

end for

end while

**DFS Pseudocode (for graph with n vertices):**

Input: grapharray[n][n], source

stack<int> S

int visited[n] (array to keep track of nodes visited, all values set to 0 except source which is set to 1 i.e. 0 = not visited, 1 = visited)

S.push(source)

while(S is not empty)

v = S.top

S.pop()

for each neighbor w of v

if visited[w] = 0

print w

visited[w] = 1

S.push(w)

end if

end for

end while

Bonus (4 points): Show *all *of the steps (splitting and merging) when using mergesort to sort (smallest to largest) the following array (they are the numbers 1 through 16):

[16, 1, 15, 2, 14, 3, 13, 4, 12, 5, 11, 6, 10, 7, 9, 8]

Bonus (2 points): describe how you could implement a queue using 2 stacks.

Bonus (4 points) Draw the binary search tree that would be constructed by inserting the following values in the exact order given (starting with an empty tree i.e. first value will be the first node in the tree): –

a. Binary Search Tree A: 8, 9, 2, 7, 1, 10, 3, 5, 6, 4

b. Binary Search Tree B: 10, 7, 9, 12, 4, 2, 5, 3, 1, 14, 11, 19, 13, 18, 20

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