Data Structures and Algorithms: cram pack

Formula sheets, the key method steps for every topic and 6 practice problems, with full solutions at the back.

New problemsBack to the course

In the print window, choose “Save as PDF” to keep a copy on your device.

1Formulas and methods

Graph theory

Handshake lemma
Σ deg(v) = 2|E|
Complete graphs
|E(Kn)| = n(n − 1)/2, |E(Km,n)| = m·n
Regular graph
|E| = n·k/2
Tree and forest
|E| = n − 1, |E| = n − c
Leaves of a tree
L = 2 + Σ (deg(v) − 2) over deg(v) ≥ 3
Euler
circuit: all degrees even; path: exactly 2 odd degrees
Hamilton circuits in Kn
(n − 1)!/2
Dijkstra update
d(v) = min(d(v), d(u) + w(u, v))
Walks of length L
(Mᴸ)ij

Method

  1. Write down the degree of every vertex; the degrees add up to twice the number of edges.
  2. For Euler questions, check that the graph is connected and count the vertices with odd degree.
  3. For trees, use n − 1 edges and the handshake lemma.
  4. For Dijkstra, keep a table of tentative distances and settle the smallest unsettled vertex each round.
  5. For a minimum spanning tree, sort the edges by weight and skip every edge that closes a cycle (Kruskal), or grow one tree from a start vertex (Prim).

2Practice problems

1.Algorithms & complexityMedium

Why do many library sorting routines switch to insertion sort for very small subarrays?

  1. AInsertion sort is O(n log n) in the worst case
  2. BInsertion sort has little overhead and is fast on small or nearly sorted inputs
  3. CInsertion sort uses less memory than any other sorting algorithm on large inputs
  4. DRecursive algorithms cannot handle arrays below 10 elements
2.Algorithms & complexityHard

Strassen's algorithm multiplies matrices using 7 instead of 8 recursive multiplications of half-size blocks. What is its running time?

  1. AΘ(n³)
  2. BΘ(n² log n)
  3. CΘ(n2)
  4. DΘ(nlog₂ 7), about n2.81
3.Data structuresMedium

Which data structure does a running program use to keep track of active function calls and their local variables?

  1. AA binary search tree of frames
  2. BThe call stack
  3. CA hash table of functions
  4. DThe heap memory queue
4.Data structuresHard

In a circular buffer, front == rear can mean either empty or full. What is a common way to tell them apart?

  1. AAlways clear the buffer whenever front equals rear, so the state is empty
  2. BKeep a separate element count, or always leave one slot unused
  3. CMake the buffer size a power of two
  4. DStore the front and rear in the same variable
5.Graph theoryMedium

A connected graph has 6 vertices and 14 edges. How many edges k must be removed to leave a spanning tree?

|V| = 6|E| = 14
k =
6.Graph theoryHard
1021061511211137ABCDEFG

Use Dijkstra’s algorithm to find the length d(F) of the shortest path from A to F in the weighted graph.

start: Aend: F
d(F) =

3Solutions

1.Algorithms & complexity

Answer: B · Insertion sort has little overhead and is fast on small or nearly sorted inputs

Why this is the answer

For small n the simple loop of insertion sort beats the recursion overhead of quicksort or merge sort, even though it is O(n²).

Answer

B · Insertion sort has little overhead and is fast on small or nearly sorted inputs
2.Algorithms & complexity

Answer: D · Θ(nlog₂ 7), about n2.81

Why this is the answer

With a = 7 and b = 2, nlog₂ 7 dominates the O(n²) addition work, giving about n2.81.

Answer

D · Θ(nlog₂ 7), about n2.81
3.Data structures

Answer: B · The call stack

Why this is the answer

Each call pushes a stack frame and each return pops it, which is why the most recent call finishes first.

Answer

B · The call stack
4.Data structures

Answer: B · Keep a separate element count, or always leave one slot unused

Why this is the answer

With a count, empty means count = 0 and full means count = capacity; with one empty slot, full is defined as (rear + 1) mod n == front.

Answer

B · Keep a separate element count, or always leave one slot unused
5.Graph theory

Answer: k = 9

Euler, Hamilton and trees: a connected graph has an Euler circuit (every edge once, back to the start) when every vertex has even degree, and an Euler path that is not a circuit when exactly two vertices have odd degree; the path then runs from one odd vertex to the other. A Hamilton circuit visits every vertex once; Kn has (n − 1)!/2 of them. A tree on n vertices has n − 1 edges, and a connected graph with m edges keeps a spanning tree after removing m − n + 1 edges.

Edges of a spanning tree

6 − 1 = 5

Edges to remove

k = 14 − 5 = 9

Answer

k = 9
6.Graph theory

Answer: d(F) = 27

Shortest paths and spanning trees: Dijkstra’s algorithm settles one vertex per round: the unsettled vertex with the smallest tentative distance. Its distance is then final, and every neighbour v gets d(v) = min(d(v), d(u) + w(u, v)). Kruskal’s algorithm builds a minimum spanning tree by taking the edges from light to heavy and skipping any edge that would close a cycle; Prim’s algorithm grows one tree from a start vertex by always adding the lightest edge that leaves it. A spanning tree of n vertices has n − 1 edges.

Dijkstra round by round

Each line: the vertex settled, its final distance, and the neighbours whose distance improves.

A (0): C: 10
C (10): B: 12, E: 16, G: 21
B (12): D: 22, G: 14
G (14)
E (16): F: 27
D (22)
F (27)

Shortest path

A → C → E → F
d(F) = 27

Answer

d(F) = 27

ExamProblems · examproblems.com/courses/data-structures-and-algorithms