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

Which statement about P and NP is correct?

  1. AEvery problem in P is also in NP
  2. BNP means "not polynomial"
  3. CNP problems cannot be verified quickly
  4. DIt has been proven that P ≠ NP
3.Data structuresMedium

What happens when a hash table rehashes after its load factor exceeds a threshold?

  1. AOnly the colliding keys are moved to a new overflow list at the end of the table
  2. BA larger table is allocated and every key is inserted again
  3. CThe oldest keys are deleted to make room
  4. DThe existing keys are sorted in place
4.Data structuresHard

Which pair of traversals always determines a unique binary tree with distinct keys?

  1. APreorder and inorder
  2. BPreorder alone
  3. CPreorder and postorder
  4. DLevel-order alone
5.Graph theoryMedium

How many different Hamilton circuits H does the complete graph K4 have? Circuits that differ only in the starting vertex or the direction count as the same.

K4
H =
6.Graph theoryHard
11141615517312ABCDEF

Kruskal’s algorithm builds a minimum spanning tree. Click the edge it adds as edge number 3; skipped edges do not count.

k = 3
  1. 1edge AB
  2. 2edge AC
  3. 3edge BC
  4. 4edge BD
  5. 5edge DE
  6. 6edge DF
  7. 7edge EF
  8. 8edge BE

3Solutions

1.Algorithms & complexity

Answer: C · 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

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

Answer: A · Every problem in P is also in NP

Why this is the answer

P ⊆ NP: if you can solve a problem quickly you can verify a solution quickly. Whether P = NP is still open.

Answer

A · Every problem in P is also in NP
3.Data structures

Answer: B · A larger table is allocated and every key is inserted again

Why this is the answer

Since the index is computed modulo the table size, all keys must be rehashed into the new, larger table; with doubling this stays O(1) amortised.

Answer

B · A larger table is allocated and every key is inserted again
4.Data structures

Answer: A · Preorder and inorder

Why this is the answer

Preorder gives the root, and its position in the inorder sequence splits the left and right subtrees; preorder with postorder is ambiguous when a node has one child.

Answer

A · Preorder and inorder
5.Graph theory

Answer: H = 3

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.

Count

H = (4 − 1)!/2 = 6/2 = 3

Answer

H = 3
6.Graph theory

Answer: edge AB

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.

Kruskal’s algorithm

1. EF (3)
2. DE (5)
3. AB (11)
4. BE (12)
5. AC (14)

edge number 3

AB

Answer

edge AB

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