What should be included in a freshman 'Mathematics for computer programmers' course?

TL;DR 2-semester course is not enough.

Disclaimer: I write this as a computer-scientist that uses math a lot in his work (I'm a research assistant at a university).

Introduction:

There are three (overlapping) aspects of math in computer science:

• Math that is actually useful.
• Math that you can run into, and is generally good to know.
• Math that lets you build more awesome math.

First is essential, because the students need to be able to do stuff. Second is important, because you cannot teach students everything, and at the same time trying to get into a new field all by yourself is quite hard (i.e. it's good to know the barest basics of everything). The third are these which aren't directly useful, but present meta-concepts that happen all the time (a bit like design patterns in programming); you can live without them, but intuition you gain there makes life much easier.

The list was sorted by (subjectively defined) importance.

Math that is actually useful.

• Logic. How to handle quantifiers and read formulas, tautologies, satisfiability; less important: free variables, substitutions and unifiers.
• Asymptotics. This is a must. The big-oh notation, etc., term manipulation, basic limits, sequences and series, both →∞→∞ and →0→0.
• Graph theory. This is a must. Graphs are so prevalent and ubiquitous nowadays that it would be a major setback to not know the basics of graph theory (e.g. searching algorithms, connectedness and flows).
• Linear algebra. This is a must. What is a linear space, matrices, determinants, orthogonalization, linear programming, perhaps even eigenvalues.
• Probability. Strongly desirable. Discrete spaces, Bayes theorem, expected values, Markov's, Chebyshev and Chernoff inequalities, the law of large numbers.
• Data processing. Strongly desirable. It's important to know how to process a gathered set of data without destroying some of its vital properties. This includes a bit of statistics, some filters, means, clustering, basic approximation, modelling and regression.

Math that you can run into, and is generally good to know.

• Numerical analysis. Floating-point arithmetic and error-analysis are a must.
• Universal algebra. It's essential to make the students not afraid of abstract structures. No advanced stuff, no need for all the groups/rings/skew fields definitions. The barest essentials so that students won't run scared when they see algebraic data types in functional programming, monoids in automata theory or group theory in cryptography.
• Statistical and probabilistic modelling. This is strongly desirable. Markov chains and Monte-Carlo methods.
• Differential equations. Differential equations happen to explain a lot about world's interactions and are used frequently in multitude of practical applications. This is strongly desirable, because it's just hard to learn them with no help.
• Information theory and cryptography. This is a must for modern programmers. Symmetric and public-key cryptography, handshakes, entropy, notion of entropy extractor, Kolmogorov complexity, impossibility of perfect compression, lower bound on comparation-based sorting complexity.
• Signal processing. Fourier transform is strongly desirable. Kalman filters would be good, but there's not enough time. Just anything that would let them reduce the noise in a practical setting.

Math that lets you build more awesome math.

• Term rewriting. Notion of term rewriting is very powerful and underappreciated. A lot of processes can be explained in terms of term rewriting and it's good to know the relevant intuitions. It's useful for showing termination property of algorithms.
• Automata theory. The notion of finite automaton is a must.
• Complexity theory. Basic complexity classes and inter-dependencies (including randomized algorithms classes). Maybe other models of computations like