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

A problem is NP-complete if it is:

  1. AImpossible to solve
  2. BSolvable in polynomial time
  3. COutside NP but harder than all NP problems, such as the halting problem
  4. DIn NP and every NP problem reduces to it in polynomial time
3.Data structuresMedium

A binary tree has root 1 with left child 2 and right child 3; node 2 has children 4 (left) and 5 (right). What is the preorder traversal?

  1. A1 2 3 4 5
  2. B4 2 5 1 3
  3. C4 5 2 3 1
  4. D1 2 4 5 3
4.Data structuresHard

For a hash table to work correctly, which rule must hash and equality follow?

  1. AHash values must be prime numbers
  2. BEqual hash values imply equal keys
  3. CDifferent keys must have different hash values
  4. DEqual keys must have equal hash values
5.Graph theoryMedium

A connected graph has 9 vertices and 15 edges. How many edges k must be removed to leave a spanning tree?

|V| = 9|E| = 15
k =
6.Graph theoryHard
811811365711ABCDEF

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: C · 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

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

Answer: D · In NP and every NP problem reduces to it in polynomial time

Why this is the answer

NP-complete problems are the hardest in NP; a polynomial algorithm for one would give one for all of NP.

Answer

D · In NP and every NP problem reduces to it in polynomial time
3.Data structures

Answer: D · 1 2 4 5 3

Why this is the answer

Preorder visits 1, then its left subtree (2, 4, 5), then 3; the other options are inorder, postorder and level-order.

Answer

D · 1 2 4 5 3
4.Data structures

Answer: D · Equal keys must have equal hash values

Why this is the answer

If two equal keys hashed differently they would be stored in different buckets and lookups would fail; unequal keys may still share a hash.

Answer

D · Equal keys must have equal hash values
5.Graph theory

Answer: k = 7

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

9 − 1 = 8

Edges to remove

k = 15 − 8 = 7

Answer

k = 7
6.Graph theory

Answer: d(F) = 14

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): B: 8, C: 11
B (8): D: 9, E: 19
D (9): E: 15, F: 14
C (11)
F (14)
E (15)

Shortest path

A → B → D → F
d(F) = 14

Answer

d(F) = 14

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