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.

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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

Which statement about a recursive function without a reachable base case is correct?

  1. AIt keeps calling itself until the call stack overflows
  2. BIt runs in O(1) time
  3. CIt returns 0 automatically
  4. DThe compiler always turns it into a loop that ends safely
2.Algorithms & complexityHard

Which technique gives an expected O(n log n) running time for quicksort on any input?

  1. AChoosing the pivot uniformly at random
  2. BSorting the input before running quicksort
  3. CUsing recursion only on the left side
  4. DAlways choosing the first element
3.Data structuresMedium

Why is traversing an array often much faster in practice than traversing a linked list of the same length?

  1. AArray elements are contiguous, so they make good use of the CPU cache
  2. BArray traversal is O(1) in total, while linked list traversal is O(n)
  3. CLinked lists always store elements on disk
  4. DArrays use fewer instructions to compare values
4.Data structuresHard

How does a radix tree (compressed trie) improve on a plain trie?

  1. AIt merges chains of single-child nodes into one edge labelled with a string
  2. BIt stores each word in a hash table at the root instead of in separate nodes
  3. CIt sorts the children of each node by frequency
  4. DIt keeps only the first letter of each word
5.Graph theoryMedium
ABCDEFEuler circuitEuler path, no circuitneither

Does the connected graph have an Euler circuit, an Euler path that is not a circuit, or neither? Click your answer.

|V| = 6|E| = 6
  1. 1Euler circuit
  2. 2Euler path, no circuit
  3. 3neither
6.Graph theoryHard
42886133811ABCDEFG

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: A · It keeps calling itself until the call stack overflows

Why this is the answer

Without a base case the recursion never stops; in practice memory for stack frames runs out and the program crashes.

Answer

A · It keeps calling itself until the call stack overflows
2.Algorithms & complexity

Answer: A · Choosing the pivot uniformly at random

Why this is the answer

With a random pivot, the expected running time is O(n log n) for every input; bad cases still exist but are very unlikely.

Answer

A · Choosing the pivot uniformly at random
3.Data structures

Answer: A · Array elements are contiguous, so they make good use of the CPU cache

Why this is the answer

Contiguous memory gives good locality of reference, while linked nodes may be scattered and cause many cache misses.

Answer

A · Array elements are contiguous, so they make good use of the CPU cache
4.Data structures

Answer: A · It merges chains of single-child nodes into one edge labelled with a string

Why this is the answer

Collapsing paths with no branching reduces the number of nodes and the memory used while keeping prefix search.

Answer

A · It merges chains of single-child nodes into one edge labelled with a string
5.Graph theory

Answer: neither

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.

Degrees

deg(A) = 2, deg(B) = 3, deg(C) = 3, deg(D) = 1, deg(E) = 2, deg(F) = 1

Odd vertices

B, C, D, F → 4
neither

Answer

neither
6.Graph theory

Answer: d(F) = 18

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: 4
B (4): C: 6, D: 12, G: 7
C (6)
G (7): E: 18
D (12): F: 18
E (18)
F (18)

Shortest path

A → B → D → F
d(F) = 18

Answer

d(F) = 18

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