Memory-hard function

In cryptography, a memory-hard function (MHF) is a function that costs significant amount of memory to evaluate. It is different from a memory-bound function; the latter incurs cost by slowing down computation through memory latency. MHFs find their use as a form of proof of work.
Memory hard measure.
There are different ways to measure the memory hardness of a function. A commonly seen measure is Cumulative Memory Complexity (CMC). In a parallel model, CMC measures memory hardness by summing up all the inputs on each step. 
Another viable measure is integrating memory against physical time.
Yet another measure is the memory bandwidth consumption on a memory bus. This category of functions are also dubbed "bandwidth-hard functions".
Motivation.
There is a reason why MHFs cost a lot of memory instead of, say, CPU cycles. Bitcoin used repeated evaluation of SHA-2 function as proof of work, but it turned out that modern general-purpose processors, i.e. off-the-shelf CPUs, are very inefficient when tasked to compute a fixed function over and over. Miners adopted application-specific integrated circuits (ASICs) and achieved 10^16 speedup. While this is fine for what Bitcoin is good for, we want a more "egalitarian" hardness measure. In other words, we want everyone to be equally inefficient in computing the function even if they have an ASIC. Because if some people can evaluate the function efficiently and some can't, then in order to make the function relatively hard for the short-cut takers, we will make the function too hard for a regular user. If everyone is inefficient, then everyone can evaluate a moderately-hard function.
Over time, it has been recognized that memory cost remains fairly equal across the board. Hence MHF.
Variants.
Based on their evaluation patterns, MHFs can be put into two camps: data-dependent (dMHF) and data-independent (iMHF). dMHFs are that which sometimes you don't know which pieces of information you would still need for later calculations, and iMHFs are ones that there's no such ambiguity. Examples of dMHFs are scrypt and Argon2d. Examples of iMHFs are Argon2i and catena. Many of these MHFs are developed to be used as password hashing functions exactly because of their memory hardness.
dMHFs have the glaring problem that they are prone to side-channel attacks like cache timing. People tend toward iMHFs for this reason, especially for password hashing. However, iMHFs are mathematically proven to have weaker memory hardness properties than dMHFs.
Construction.
depth-robust graph.
For iMHFs in the parallel random oracle model (pROM), it is a known fact that the cumulative pebbling complexity is lower-bounded and upper-bounded by the depth-robustness of a graph.